Advertisement

Spatial Statistics and Micromechanics of Materials

  • Dominique Jeulin
Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 600)

Abstract

The prediction of elastic properties and of fracture strength of materials from their microstructure is an old but still active problem, implying many theoretical questions and important practical consequences. In both cases, statistical correlation functions deduced from the geometry by image analysis measurements, or obtained for specific models of random media, enter into the estimation of overall properties of materials. Concerning elastic properties of random media, useful bounds can be worked out from their three-points statistics. This is illustrated for various models of random media, including multiscale textures producing so-called “optimal microstructures”. This method also provides estimators (upper bounds) of the elastic moduli of porous media. Fracture statistics models based on random functions allow us to predict the probability of fracture of materials, depending on the scale, the applied stress field, and the microstructure. Recent developments are reviewed in brittle fracture (using the weakest link model, or Griffith crack propagation in random media) and in the case of materials submitted to an increasing damage during the fracture process.

Keywords

Duplex Stainless Steel Fracture Statistic Boolean Model Hard Sphere Model Order Bound 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arns, C., M. Knackstedt, K. Mecke (2002): ‘Characterising the morphology of disordered materials’, this volume.Google Scholar
  2. 2.
    Barbe, F., L. Decker., D. Jeulin, G. Cailletaud (2001): ‘Intergranular and intragranular behavior of polycristalline aggregates. Part 1: F.E. model’, Int. J. Plasticity 17(4), pp. 513–536.zbMATHCrossRefGoogle Scholar
  3. 3.
    Baxevanakis, C., D. Jeulin, J. Renard (1995): ‘Fracture Statistics of a Unidirectional Composite’, Int. J. Fracture 73, pp. 149–181.CrossRefGoogle Scholar
  4. 4.
    Baxevanakis, C., D. Jeulin, B. Lebon, J. Renard (1998): ‘Fracture statistics modeling of laminate composites’, Int. J. Solids Struct. 35, No 19, pp. 2505–2521.zbMATHCrossRefGoogle Scholar
  5. 5.
    Beran, M.J., J. Molyneux (1966): ‘Use of classical variational principles to determine bounds for the effective bulk modulus in heterogeneous media’, Q. Appl. Math. 24, pp. 107–118.zbMATHGoogle Scholar
  6. 6.
    Beran, M. J. (1968): Statistical Continuum Theories. (J. Wiley, New York).zbMATHGoogle Scholar
  7. 7.
    Berdin, C., G. Cailletaud, D. Jeulin (1993): ‘Micro-Macro Identification of Fracture Probabilistic Models’. In: Proc. of the International Seminar on Micromechanics of Materials, MECAMAT’93, Fontainebleau, 6–8 July 1993 (Eyrolles, Paris), pp. 499–510.Google Scholar
  8. 8.
    Beremin, F.M. (1983): ‘A local criterion for cleavage fracture of a nuclear pressure vessel steel’, Metall. Trans. A. 14A, pp. 2277–2287.ADSGoogle Scholar
  9. 9.
    Bergman, D. (1978): ‘The dielectric constant of a composite material: a problem in classical physics’, Phys. Rep. C 43, pp. 377–407.CrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Berryman, G.J. (1985): ‘Variational bounds on elastic constants for the penetrable sphere model’, J. Phys. D: Appl. Phys. 18, pp. 585–597.CrossRefADSGoogle Scholar
  11. 11.
    Berryman, G.J., G.W. Milton (1988): ‘Microgeometry of random composites and porous media’, J. Phys. D: Appl. Phys. 21, pp. 87–94.CrossRefADSGoogle Scholar
  12. 12.
    Bretheau, T., D. Jeulin (1989): ‘Caractéristiques morphologiques des constituants et comportement à la limite éelastique d’un matériau biphasé Fe/Ag’, Rev. Phys. Appl. 24, pp. 861–869.Google Scholar
  13. 13.
    Cailletaud, G., D. Jeulin, Ph. Rolland (1994): ‘Size effect on elastic properties of random composites’, Eng. Comput. 11(2), pp. 99–110.CrossRefGoogle Scholar
  14. 14.
    Chudnovsky, A., B. Kunin (1987): ‘A probabilistic model of brittle crack formation’, J. Appl. Phys. 62(10), pp. 4124, 4129.CrossRefADSGoogle Scholar
  15. 15.
    Daniels, H.E. (1945): ‘The statistical theory of the Strength of Bundles of Threads’, Proc. Roy. Soc. London A 183, pp. 405–435.zbMATHADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Decamp, K., L. Bauvineau, J. Besson, A. Pineau (1997): ‘Size and geometry effect on ductile fracture of notched bars in a C-Mn steel: experiments and modelling’, Int. J. Fracture 88, pp. 1–18.CrossRefGoogle Scholar
  17. 17.
    Decker, L., D. Jeulin (2000): ‘Simulation 3D de matériaux aléatoires polycristallins’, Revue de Métallurgie-CIT/Science et Génie des Matériaux, Feb. 2000, pp. 271–275.Google Scholar
  18. 18.
    Delarue, A., D. Jeulin, in preparation.Google Scholar
  19. 19.
    Delisée, Ch., D. Jeulin, F. Michaud (2001): ‘Caractérisation morphologique et porosité en 3D de matériaux fibreux cellulosiques’, C.R. Acad. Sci. Paris, vol. 329, Série II b, pp. 179–185.Google Scholar
  20. 20.
    Devillers-Guerville, L., J. Besson, A. Pineau (1997): ‘Notch fracture toughness of a cast duplex stainless steel: modeling of experimental scatter and size effect’, Nucl. Eng. Design 168, pp. 211–225.CrossRefGoogle Scholar
  21. 21.
    Eyre, D.J., G.W. Milton (1999): ‘A fast numerical scheme for computing the response of composites using grid refinement’, Eur. Phys. J. Appl. Phys. 6, pp. 41–47.CrossRefADSGoogle Scholar
  22. 22.
    Harter, H.L. (1978): ‘A bibliography of extreme-value theory’, Int. Stat. Rev. 46, pp. 279–306.zbMATHMathSciNetGoogle Scholar
  23. 23.
    Hashin, Z., S. Shtrikman (1962): ‘A variational approach to the theory of the effective magnetic permeability of multiphase materials’, J. Appl. Phys. 33, pp. 3125–3131.zbMATHCrossRefADSGoogle Scholar
  24. 24.
    Hashin, Z., S. Shtrikman (1963): ‘A variational approach to the theory of the elastic behavior of multiphase materials’, J. Mech. Phys. Solids 11, pp. 127–140.CrossRefADSMathSciNetzbMATHGoogle Scholar
  25. 25.
    Heijmans H.J.A.M. (1994): Morphological Image Operators. Advances in Electronics and Electron Physics. Vol. Suppl. 24. (Academic Press, Boston).Google Scholar
  26. 26.
    Hori, M. (1973): ‘Statistical theory of the effective electrical, thermal, and magnetic properties of random heterogeneous materials. II. Bounds for the effective permittivity of statistically anisotropic materials’, J. Math. Phys. 14, pp. 1942–1948.CrossRefADSGoogle Scholar
  27. 27.
    Hug, D., G. Last, W. Weil (2002): ‘A Survey on Contact Distributions’, this volume.Google Scholar
  28. 28.
    Jeulin, D. (1978): ‘Amélioration des performances des hauts fourneaux: recherche des relations entre morphologie et qualité des agglomérés’, IRSID Report RE 578, Rapport final CECA (convention no 6 220. AA/3/304).Google Scholar
  29. 29.
    Jeulin, D. (1981): ‘Improvement of blast furnace performance: research into relations between morphology and quality of sinters’, E.E.C. Publication EUR 6280.Google Scholar
  30. 30.
    Jeulin, D., P. Jeulin (1981): ‘Synthesis of Rough Surfaces by Random Morphological Models’. In: Proc. 3rd European Symposium of Stereology, Stereol. Iugosl. 3,suppl. 1, pp. 239–246.Google Scholar
  31. 31.
    Jeulin, D. (1986): ‘Study of spatial distributions in multicomponent structures by image analysis’. In: Proc. 4th European Symposium of Stereology (Göteborg, 26–30 August 1985), Acta Stereol. 5/2, pp. 233–239.Google Scholar
  32. 32.
    Jeulin, D. (1987): ‘Random structure analysis and modelling by Mathematical Morphology’. In: Proc. CMDS5, ed. by A. J. M. Spencer (Balkema, Rotterdam), pp. 217–226.Google Scholar
  33. 33.
    Jeulin, D. (1991): Modèles morphologiques de structures aléatoires et de changement d’échelle, Thèse de Doctorat d’Etat, University of Caen.Google Scholar
  34. 34.
    Jeulin, D. (1992): ‘Some Crack Propagation Models in Random Media’. In: Proc. Symposium on the Macroscopic Behavior of the Heterogeneous Materials from the Microstructure, ASME, Anaheim, Nov 8–13, 1992. AMD Vo. 147, pp. 161–170.Google Scholar
  35. 35.
    Jeulin, D. (1994): ‘Random structure models for composite media and fracture statistics’. In: Advances in Mathematical Modelling of Composite Materials, ed. by K.Z. Markov (World Scientific Company, Singapore), pp. 239–289.Google Scholar
  36. 36.
    Jeulin, D., C. Baxevanakis, J. Renard (1995): ‘Statistical modelling of the fracture of laminate composites’. In: Applications of Statistics and Probability, ed. by M. Lemaire, J.L. Favre, A. Mébarki (Balkema, Rotterdam), pp. 203–208.Google Scholar
  37. 37.
    Jeulin, D., A. Le Coënt (1996): ‘Morphological modeling of random composites’, Proceedings of the CMDS8 Conference (Varna, 11–16 June 1995), ed. by K.Z. Markov (World Scientific, Singapore), pp. 199–206.Google Scholar
  38. 38.
    Jeulin, D. (ed) (1997) Proceedings of the Symposium on the Advances in the Theory and Applications of Random Sets (Fontainebleau, 9–11 October 1996) (World Scientific, Singapore).Google Scholar
  39. 39.
    Jeulin, D., L. Savary (1997): ‘Effective Complex Permittivity of Random Composites’, J. Phys. I/ Condens.Matter 7, pp. 1123–1142.Google Scholar
  40. 40.
    Jeulin, D. (1998): ‘Bounds of physical properties of some random structure’. In: Proceedings of the CMDS9 Conference (Istanbul, Turkey, June 29–July 3, 1998), ed. by E. Inan and K.Z. Markov (World Scientific, Singapore), pp. 147–154.Google Scholar
  41. 41.
    Jeulin, D. (2000): ‘Models of random damage’. In: Proc.Euromat 2000 Conference, Tours, France, ed. by D. Miannay, P. Costa, D. François, A. Pineau, pp. 771–776.Google Scholar
  42. 42.
    Jeulin, D., P. Monnaie, F. Péronnet (2001): ‘Gypsum morphological analysis and modeling’, Cement and Concrete Composites 23(2–3), pp. 299–311.CrossRefGoogle Scholar
  43. 43.
    Jeulin, D. (2001): ‘Random Structure Models for Homogenization and Fracture Statistics’, In: Mechanics of Random and Multiscale Microstructures, ed. by D. Jeulin, M. Ostoja-Starzewski (CISM Lecture Notes No 430, Springer Verlag).Google Scholar
  44. 44.
    Jeulin, D., M. Ostoja-Starzewski (eds) (2001): Mechanics of Random and Multiscale Microstructures. (CISM Lecture Notes No 430, Springer Verlag).Google Scholar
  45. 45.
    Jeulin, D.: Morphological Models of Random Structures, in preparation.Google Scholar
  46. 46.
    Kittl, P., G. Diaz (1986): ‘Weibull’s fracture statistics, or probabilistic strength of materials: state of the art’, Res. Mechanica 25, pp. 99–207.Google Scholar
  47. 47.
    Kröner, E. (1971): Statistical Continuum Mechanics. (Springer Verlag, Berlin).Google Scholar
  48. 48.
    Le Coënt, A., D. Jeulin (1996): ‘Bounds of effective physical properties for random polygons composites’, C.R. Acad. Sci. Paris, 323, Série II b, pp. 299–306.zbMATHGoogle Scholar
  49. 49.
    Matheron, G. (1967): Eléments pour une théorie des milieux poreux. (Masson, Paris).Google Scholar
  50. 50.
    Matheron, G. (1968): ‘Composition des perméabilités en milieu poreux hétérogène: critique de la règle de pondération géométrique’, Rev. IFP 23, pp. 201–218.Google Scholar
  51. 51.
    Matheron, G. (1971): The theory of regionalized variables and its applications. (Paris School of Mines publication).Google Scholar
  52. 52.
    Matheron, G. (1975): Random sets and integral geometry. (J. Wiley, New York).zbMATHGoogle Scholar
  53. 53.
    McCoy, J.J. (1970): ‘On the deplacement field in an elastic medium with random variations of material properties’. In: Recent Advances in Engineering Sciences, Vol. 5 (Gordon and Breach, New York), pp. 235–254.Google Scholar
  54. 54.
    Mecke, K., H. Wagner (1991): ‘Euler characteristics and related measures for random geometric sets’, J. Statist. Phys. 64, pp. 843–850.zbMATHCrossRefADSMathSciNetGoogle Scholar
  55. 55.
    Mecke, K. (2000): ‘Additivity, convexity, and beyond: Applications of Minkowski functionals in statistical physics’. In: Statistical Physics and Spatial Statistics, ed. by K.R. Mecke, D. Stoyan, Lecture Notes in Physics 554 (Springer Verlag, Berlin), pp. 111–184.CrossRefGoogle Scholar
  56. 56.
    Miller, M.N. (1969): ‘Bounds for the effective electrical, thermal and magnetic properties of heterogeneous materials’, J. Math. Phys. 10, pp. 1988–2004.CrossRefADSGoogle Scholar
  57. 57.
    Miller, M.N. (1969): ‘Bounds for effective bulk modulus of heterogeneous materials’, J. Math. Phys. 10, pp. 2005–2013.CrossRefADSGoogle Scholar
  58. 58.
    Milton, G. (1980): ‘Bounds on the complex dielectric constant of a composite material’, Appl. Phys. Lett. 37, pp. 300–302.CrossRefADSGoogle Scholar
  59. 59.
    Milton, G. (1982): ‘Bounds on the elastic and transport properties of two component composites’, J. Mech. Phys. Solids 30, pp. 177–191.zbMATHCrossRefADSMathSciNetGoogle Scholar
  60. 60.
    Molchanov, I. (1997): Statistics of the Boolean model for practitioners and mathematicians. (J. Wiley, Chichester).zbMATHGoogle Scholar
  61. 61.
    Moulinec H., P. Suquet (1994): ‘A fast numerical method for computing the linear and nonlinear mechanical properties of composites’, C.R. Acad. Sci. Paris, 318, Série II, pp. 1417–1423.zbMATHGoogle Scholar
  62. 62.
    Pélissonnier-Grosjean, C., D. Jeulin, L. Pottier, D. Fournier, A. Thorel (1997): ‘Mesoscopic modeling of the intergranular structure of Y2O3 doped aluminium nitride and application to the prediction of the effective thermal conductivity’. In: Key Engineering Materials, Volumes 132–136, Part 1 (Transtech Publications, Switzerland), pp. 623–626.Google Scholar
  63. 63.
    Ponte Castaneda, P. (1996): ‘Variational methods for estimating the effective behavior of nonlinear composite materials’. In: Proceedings of the CMDS8 Conference (Varna, 11–16 June 1995), ed. by K.Z. Markov (World Scientific, Singapore), pp. 268–279.Google Scholar
  64. 64.
    Quenec’h, J.L., J.L. Chermant, M. Coster, D. Jeulin (1994): ‘Liquid phase sintered materials modelling by random closed sets’. In: Mathematical morphology and its applications to image processing, ed. by J. Serra, P. Soille (Kluwer, Dordrecht), pp. 225–232.Google Scholar
  65. 65.
    Rintoul, M.D., S. Torquato (1997): ‘Precise determination of the critical threshold and exponents in a three-dimensional continuum percolation model’, J. Phys. A: Math. Gen. 30, pp. L585–L592.CrossRefADSGoogle Scholar
  66. 66.
    Serra, J. (1982): Image analysis and mathematical morphology. (Academic Press, London).zbMATHGoogle Scholar
  67. 67.
    Soille, P. (1999): Morphological Image Analysis: Principles and Applications. (Springer, Berlin).zbMATHGoogle Scholar
  68. 68.
    Silnutzer, N.R. (1972): Effective constants of statistically homogeneous materials, PhD thesis, University of Pennsylvania.Google Scholar
  69. 69.
    Stoyan, D., W.S. Kendall, J. Mecke (1987): Stochastic Geometry and its Applications. (J. Wiley, New York).zbMATHGoogle Scholar
  70. 70.
    Suh, M.W., B.B. Bhattacharyya, A. Grandage (1970): ‘On the distribution and moments of the strength of a bundle of filaments’, J. Appl. Prob. 7, pp. 712–720.zbMATHCrossRefMathSciNetGoogle Scholar
  71. 71.
    Torquato, S., G. Stell (1983): ‘Microstructure of two-phase random media III. The n-point matrix probability functions for fully penetrable spheres’, J. Chem. Phys. 79, pp. 1505–1510.CrossRefADSMathSciNetGoogle Scholar
  72. 72.
    Torquato, S., F. Lado (1986): ‘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, pp. 6428–6434.CrossRefADSGoogle Scholar
  73. 73.
    Torquato, S. (1991): ‘Random heterogeneous media: microstructure and improved bounds on effective properties’, Appl. Mech. Rev. 44, pp. 37–76.MathSciNetCrossRefGoogle Scholar
  74. 74.
    Vogel, H.J. (2002): ‘Topological characterization of porous media’, this volume.Google Scholar
  75. 75.
    Willis, J.R. (1981): ‘Variational and related methods for the overall properties of composites’. In: Advances in Applied Mechanics, 21, pp. 1–78.zbMATHMathSciNetCrossRefGoogle Scholar
  76. 76.
    Willis, J.R. (1991): ‘On methods for bounding the overall properties of nonlinear composites’, J. Mech. Phys. Solids 39,1, pp. 73–86.zbMATHCrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Dominique Jeulin
    • 1
  1. 1.Centre de Morphologie Mathématique Ecole des Mines de ParisFontainebleauFrance

Personalised recommendations