SpringerLink
Log in

Statistical Description of Composite Materials

  • Chapter
  • First Online:
Local and Nonlocal Micromechanics of Heterogeneous Materials

  • 958 Accesses

Abstract

The quantitative description of the microtopology of heterogeneous media, such as composite materials, porous and cracked solids, suspensions, and amorphous materials, is crucial in the prediction of overall mechanical and physical properties of these materials. In composite materials, internal structure refers to the size and shape of inhomogeneities, defects, and crystallites (grains); to the distribution of their orientations (textures); and to spatial correlations between these mechanical, geometrical, and crystallographic features. In this chapter the basic concepts and hypotheses associated with quantification of microstructure morphology of CM are introduced in a quite general form applicable to a wide class of both random and deterministic structures. Finally, the methods of stochastic simulation of real random structures of fiber composites and particle ones are considered.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
EUR 29.95
Price includes VAT (Germany)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
EUR 213.99
Price includes VAT (Germany)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
EUR 267.49
Price includes VAT (Germany)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info
Hardcover Book
EUR 267.49
Price includes VAT (Germany)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Adams BL, Morris PR, Wang TT, Willden KS, Wright SI (1987) Description of orientation coherence in polycrystalline materials. Acta Metall, 35:2935–2946

    Article  Google Scholar 

  2. Adams BL, Olson T (1998) The metho structure – properties linkage in polycrystals. Prog Mater Sci, 43:1–88

    Article  Google Scholar 

  3. Alcaraz AN, Duhau RS, Fernández JR, Harrowell P, Miracle DB (2008) Dense amorphous packing of binary hard sphere mixtures with chemical order: The stability of a solute ordered approximation. J. Non-Crystalline Solids, 54:3171–3178

    Article  Google Scholar 

  4. Allen MP, Tildesley DJ (1987) Computer Simulations of Liquids, Oxford Science Publications, Oxford

    MATH  Google Scholar 

  5. Amparano, FE, **b Y, Roh Y-S (2000) Experimental study on the effect of aggregate content on fracture behavior of concrete. Engineering Fracture Mechanics, 67:65–84

    Article  Google Scholar 

  6. Armington M (1991) Limit distributions of the states and inhomogenization in random media. Acta Mechan, 88:27–59

    Article  Google Scholar 

  7. Aste T, Saadatfar M, Sakellariou A, Senden TJ (2004) Investigating the geometrical structure of disordered sphere packings. Physica, A339:16–23

    Article  MathSciNet  Google Scholar 

  8. Aste T, Saadatfar M, Senden TJ (2005) Geometrical structure of disordered sphere packings. Phys. Rev., E71:061302

    Google Scholar 

  9. Attard, P (1989) Spherically inhomogeneous fluids. II. Hard-sphere solute in a hard-sphere solvent. J. Chem. Phys., 91:3083–3089

    Article  Google Scholar 

  10. Baer MR, Hall CA, Gustavsen RL, Hooks DE, Sheffield SA (2007) Isentropic loading experiments of a plastic bonded explosive and constituents. J. Appl. Phys., 101:034906

    Article  Google Scholar 

  11. Banerjee B, Adams DO (2004) On predicting the effective elastic properties of polymer bonded explosives using the recursive cell method Int. J. Solids Structures, 41:481–509

    Google Scholar 

  12. Barker GC (1993) Computer simulation of granular materials. In: Mehta A (ed) Granular Matter - An Interdisciplinary Approach. Springer Verlag, Berlin

    Google Scholar 

  13. Bargmann S, Klusemann B, Markmann J, Schnabel JE, Schneider K, Soyarslan C, Wilmers J (2018) Generation of 3D representative volume elements for heterogeneous materials: A review. Progress in Materials Science, 96:322–384

    Article  Google Scholar 

  14. Barrioa C, Solana JR (2005) Map** a hard-sphere fluid mixture onto a single component hard-sphere fluid. Physica, A 351:387–403

    Google Scholar 

  15. Batchelor GK (1972) Sedimentation in a dilute dispersion of spheres. J Fluid Mech, 52:245–268

    Article  MATH  Google Scholar 

  16. Batchelor GK (1974) Transport properties of two-phase materials with random structure. Ann-Rev Fluid Mech, 6:227–255

    Article  MATH  Google Scholar 

  17. Baus M, Colot JL (1987) Thermodynamics and structure of a fluid of hard rods, disks, spheres, of hyperspheres from rescaled virial expansion. Phys Rev, A36:3912–3925

    Article  Google Scholar 

  18. Bennet CH (1972) Serially deposited amorphous aggregates of hard spheres. J Appl Phys, 43:2727–2734

    Article  Google Scholar 

  19. Beran MJ (1968) Statistical Continuum Theories. John Wiley & Sons, New York

    Book  MATH  Google Scholar 

  20. Berryman JG (1983) Random close packing of hard spheres and disks. Phys Rev, A 27:1053–1061

    Google Scholar 

  21. Berryman JG (1985) Measurement of spatial correlation functions using image processing techniques. J Appl Phys, 57:2374–2384

    Article  Google Scholar 

  22. Bhattacharyya A, Lagoudas DC (2000) Effective elastic moduli of two-phase transversely isotropic composites with aligned clustered fibers. Acta Mechan, 145:65–93

    Article  MATH  Google Scholar 

  23. Bildstein B, Kahl G (1994) Triplet correlation functions for hard-spheres: computer simulation results. J Chem Phys, 100:5882–5893

    Article  Google Scholar 

  24. Binder K, Heerman DW (1997) Monte Carlo Simulation in Statistical Physics: an Introduction. Springer, Berlin NY

    Book  Google Scholar 

  25. van Blaaderen A, Wiltzius P (1995) Real-space structure of colloidal hard-sphere glasses. Science, 270(5239), 1177–1179

    Article  Google Scholar 

  26. Boudreaux DS, Gregor JM (1977) Structure simulation of transition-metal-metalloid glasses. J Appl Phys, 48:152–158

    Article  Google Scholar 

  27. Brechet YJM (1994) Clusters, plasticity and damage: a missing link? Mater Sci Engng, A175:63–69

    Article  Google Scholar 

  28. Buevich YA (1992) Heat and mass transfer in disperse media. I. Average field equations. Int J Heat Mass Transfer, 35:2445–2452

    Article  MATH  Google Scholar 

  29. Bunge HJ (1982) Texture Analysis in Mater Science. Butterworths, Boston

    Google Scholar 

  30. Buryachenko VA (2007b) Micromechanics of Heterogeneous Materials. Springer, NY

    Book  MATH  Google Scholar 

  31. Buryachenko V (2012) Modeling of random bimodal structures of composites (application to solid propellant) II. Estimation of effective elastic moduli. Comput. Model. Engng & Sciences (CMES), 85(5). 417–446.

    Google Scholar 

  32. Buryachenko V, Jackson T, Amadio G (2012) Modeling of random bimodal structures of composites (application to solid propellant) I. Simulation of random packs. Comput. Model. Engng & Sciences (CMES), 85(5), 379–416.

    Google Scholar 

  33. Buryachenko VA, Pagano NJ, Kim RY, Spowart JE (2003) Quantitative description of random microstructures of composites and their effective elastic moduli. Int J Solids Struct, 40:47–72

    Article  MATH  Google Scholar 

  34. Cargill III GS (1994) Random packing for amorphous binary alloys. J Phys Chem Solids, 55:1375–1380

    Article  Google Scholar 

  35. Cemlins A (1988) Representation of two-phase flows by volume averaging. Int J Multiphase Flow, 14:81-91

    Article  Google Scholar 

  36. Cesarano III J, McEuen MJ, Swiler T (1995) Computer simulation of particle packing. Intern SAMPE Technical Conf, 27:658–665

    Google Scholar 

  37. Chen X, Papathanasiou TD (2004) Interface stress distributions in transversely loaded continuous fiber composites: parallel computation in multi-fiber RVEs using the boundary element method. Comp Sci Technol, 64:1101–1114

    Article  Google Scholar 

  38. Cheng YF, Guo SJ, Lay HY (2000) Dynamic simulation of random packing of spherical particles. Powder Technol, 107:123–130

    Article  Google Scholar 

  39. Chong JS, Christiansen EB, Baer AD (1971) Rheology of concentrated suspensions. J. Applied Polymer Science, 15:2007–2021

    Article  Google Scholar 

  40. Clarke AS, Willey JD (1987) Numerical simulation of the dense random packing of a binary mixture of hard spheres: amorphous metals. Phys Rev, B35:7350–7356

    Article  Google Scholar 

  41. Corson PB (1974) Correlation function for predicting properties of heterogeneous materials. I. Experimental measurement of spatial correlation functions in multiphase solids. J Appl Phys, 45:3159–3164

    Article  Google Scholar 

  42. Cox DR, Isham V (1980) Point Processes. Chapman and Hall, London and NY

    MATH  Google Scholar 

  43. Dao M, Gu P, Maewal A, Asaro RJ (1997) A micromechanical study of residual stresses in functionally graded materials. Acta Mater, 45:3265–3276

    Article  Google Scholar 

  44. Davis IL, Carter RG (1989) Random particle packing by reduced dimension algorithms. J Appl Phys, 67:1022–1029

    Article  Google Scholar 

  45. Diggle PJ (2003) Statistical Analysis of Spatial Point Patterns (2nd edn). Academic Press, New York

    MATH  Google Scholar 

  46. Döge G (2000) Grand canonical simulation of hard-disc systems by simulated tempering. In: Mecke KR, Stoyan D (eds) Statistical Physics and Spatial Statistics: The Art of Analyzing and Modeling Spatial Structures and Pattern Formation. Lecture Nnotes in Physics, Vol. 554, Berlin

    Google Scholar 

  47. Donev A, Torquato S, Stillinger FH (2005) Pair correlation function characteristics of nearly jammed disordered and ordered hard-sphere packings. J. of Computational Physics, 202:737–764

    Article  MathSciNet  MATH  Google Scholar 

  48. Enikolopyan NS, Fridman ML, Stalnova IO, Popov VL (1990) Filled polymers: mechanical properties and processability. Adv Polym Sci, 96:1–67

    Article  Google Scholar 

  49. Feder J (1980) Random sequential adsorption J Theor Biol, 87:237–254

    Google Scholar 

  50. Ferrante FJ, Arwade SR, Graham-Brady LL (2005) A translation model for non-stationary, non-Gaussian random processes. Probabilistic Engin Mech, 20:215–228

    Article  Google Scholar 

  51. Francqueville F, Gilormini P, Diani J (2019) Representative volume elements for the simulation of isotropic composites highly lled with monosized spheres - Int. J. Solids and Structures, 158:277–286

    Google Scholar 

  52. Franciosi P, Lebail H (2004) Anisotropy features of phase and particle spatial pair distributions in various matrix/inclusions structures. Acta Mater, 52:3161–3172

    Article  Google Scholar 

  53. Frisch HL (1965) Statistics of random media. Trans Soc Rheol, 9:293–312

    Article  Google Scholar 

  54. Furukawa K, Imai K, Kurashige M (2000) Simulated effect of box size and wall on porosity of random packing of spherical particles. Acta Mechan, 140:219–231

    Article  MATH  Google Scholar 

  55. Gajdošík J, Zeman J, Sejnoha M (2006) Qualitative analysis of fiber composite microstructure Influence of boundary conditions. Probabilistic Engineering Mechanics, 21:317–329

    Article  Google Scholar 

  56. Gallier S (2009) A stochastic pocket model for aluminum agglomeration in solid propellants. Propellants Explos. Pyrotech., 34:97–105

    Article  Google Scholar 

  57. Garboczi EJ, Bentz DP (1997) Analytical formulas for interfacial transition zone properties. Advanced Cement Based Materials, 6:99-108

    Article  Google Scholar 

  58. Gavrikov VL, Stoyan D (1995) The use of marked point processes in ecological and environmental forest studies. Environmental and Ecological Statistics, 2:331–344

    Article  Google Scholar 

  59. Gel’fand IM, Milos RA, Shapiro SY (1963) Representations of the Rotation and Lorentz Groups and Their Applications. Pergamon, Oxford

    Google Scholar 

  60. Ghosh S (2011) Micromechanical Analysis and Multi-Scale Modeling Using the Voronoi Cell Finite Element Method (Computational Mechanics and Applied Analysis). CRC Press, Boca Raton

    MATH  Google Scholar 

  61. Ghosh S, Moorthy S (2004) Three dimensional Voronoi cell finite element model for microstructures with ellipsoidal heterogeneties. Comput Mech, 34:510–531

    Article  MATH  Google Scholar 

  62. Ghosh S, Mukhopadhyay S N (1991) A two-dimensional automatic mesh generator for finite element analysis for random composites. Compos & Struct, 41:245–256

    Article  MATH  Google Scholar 

  63. Ghosh S, Nowak Z, Lee K (1997) Quantitative characterization and modeling of composite microstructures by Voronoi cells. Acta Mater, 45:2215–2234

    Article  Google Scholar 

  64. Gidaspow D, Huilin L (1998) Equation of state and radial distribution functions of FCC particles in a CFB. AICHE J., 44:279–293

    Article  Google Scholar 

  65. Gonzalez A, Roman FL, White JA (1999) A test-particle method for the calculation of the three-particle distribution function of the hard-sphere fluid: density functional theory and simulation. J Phys: Condens Matter, 11:3789–3998

    Google Scholar 

  66. Graham-Brady LL, Siragy EF, Baxter SC (2003) Analysis of heterogeneous composites based on moving-window techniques. J Engng Mech, 129:1054–1064

    Google Scholar 

  67. Green PI, Sibson R (1977) Computing Dirichlet tesselations in the plane. Computer J, 21:168–173

    Article  MATH  Google Scholar 

  68. Grujicic M, Zhag Y (1998) Determination of effective elastic properties of functionally graded materials using Voronoi cell finite element method. Mater Sci Engng, A251: 64–76

    Article  Google Scholar 

  69. Hahn UA, Micheletti R, Pohlink R, Stoyan D, Wendrock H (1999) Stereological analysis and modelling of gradient structures. J Microsc, 195:113–124

    Article  Google Scholar 

  70. Hall P (1988) Introduction to the Theory of Coverage Processes. John Willey & Sons, NY

    MATH  Google Scholar 

  71. Hansen JP, McDonald IR (1986) Theory of Simple Liquids. Academic Press, New York

    MATH  Google Scholar 

  72. He D, Ekere NN (2001) Structure simulation of concentrated suspensions of hard spherical particles AIChE J, 47:53–59

    Google Scholar 

  73. He D, Ekere NN, Cai L (1999) Computer simulation of random packing of unequal particles. Phys Rev, E60:7098

    Google Scholar 

  74. Henderson D, Plischke M (1987) Sum rules for the pair-correlation function of inhomogeneous fluid: results for the hard-sphere-hard-wall system. Proc Roy Soc Lond, A410:409–420

    MATH  Google Scholar 

  75. Herczynski R, Pienkowska I (1980) Toward a statistical theory of suspension. Annu Rev Fluid Mech, 12:137–269

    Article  MATH  Google Scholar 

  76. Hinch EJ (1977) An averaged-equation approach to particle interactions in a fluid suspension. J Fluid Mech, 83:695–720

    Article  MATH  Google Scholar 

  77. Hinrichsen E L, Feder J, Jossang T (1986) Geometry of random sequential adsorption. J Statist Phys, 44:793–827

    Article  Google Scholar 

  78. Illian J, Penttinen A, Stoyan H, Stoyan D (2008) Statistical Analysis and Modeling of Spatial Point Patterns. Willey & Sons, Chichester.

    Google Scholar 

  79. Jensen EBV (1998) Local Stereology. World Science, Singapore

    Book  MATH  Google Scholar 

  80. Jerier J-F, Richefeu V, Imbault D, Donzé FV (2010) Packing spherical discrete elements for large scale simulations. Comput. Methods Applied Mech. Engng, 199:1668–1676

    Article  MATH  Google Scholar 

  81. Jeulin D (2001) Random structure models for homogenization and fracture satistics. In: Jeulin D, Ostoja-Starzewski M (eds), Mechanics of random and multiscale microstructures, Springer, Wien-New York, 33–91

    Chapter  MATH  Google Scholar 

  82. Jodrey WS, Tory M (1985) Computer simulation of close random packing of equal spheres. Phys Rev, A32:2347

    Article  Google Scholar 

  83. Kanaun SK (1990) Self-consistent averaging schemes in the mechanics of matrix composite materials. Mekhanika Kompozitnikh Materialov, 26:702–711 (In Russian. Engl Transl. Mech Compos Mater, 26:984–992)

    Google Scholar 

  84. Kansal AR, Truskett TM, Torquato S (2000) Nonequilibrium hard-disk packing with controlled orientational order. J Chem Phys, 113:4844–4851

    Article  Google Scholar 

  85. Kanuparthi S, Subbarayan G, Siegmund T, Sammakia B (2009) The effect of polydispersivity on the thermal conductivity of particulate thermal interface materials. IEEE Trans. Components Packaging Technologies, 32:424–434

    Article  Google Scholar 

  86. Karlsson LM, Liljeborg A (1994) Second-order stereology for pores in translucent alumina studied by confocal scanning laser microscopy. J Microsc, 175:186–194

    Article  Google Scholar 

  87. Kim JC, Auh KH, Martin DM (2000) Multi-level particle packing model of ceramic agglomerates. Model Simul Mater Sci Engng, 8:159–168

    Article  Google Scholar 

  88. Kirkwood JG (1935) Statistical mechanics of fluid mixtures. J Chem Phys, 3:300–313

    Article  MATH  Google Scholar 

  89. Knott GM, Jackson TL, Buckmaster J (2001) Random packing of heterogeneous propellants. AIAA J, 39:678–686

    Article  Google Scholar 

  90. Kochevets S, Buckmaster J, Jackson TL, Hegab A (2001) Random propellant packs and the flames they support. AIAA J Propul Power, vol. 17:883–891

    Google Scholar 

  91. Kondrachuk AV, Shapovalov GG, Kartuzov VV (1997) Simulation modeling of the randomly nonuniform structure of powders. Two-dimensional formulation of the problem. Poroshkovaya Metallurgiya, (1-2):111-118 (In Russian. Engl Translation. Powder Metall Metal Ceram, 36:101–106)

    Google Scholar 

  92. König D, Carvajal-Gonzalz S, Downs AM, Vassy J, Rigaut JP (1991) Modelling and analysis of 3-D arrangements of particles by point process with examples of application to biological data obtained by confocal scanning light microscopy. J Microscopy, 161:405–433

    Article  Google Scholar 

  93. Kotlarchyk M, Chen S-H (1983) Analysis of small angle neutron scattering spectra from polydisperse interacting colloids. J. Chem. Phys., vol. 79, 2461–2469

    Article  Google Scholar 

  94. Kröner E (1972) Statistical Continuum Mechanics. Springer-Verlag, Vienna–New York

    MATH  Google Scholar 

  95. Kröner E (1977) Bounds for effective moduli of disordered materials. J Mech Phys Solids, 25:137–155

    Article  MATH  Google Scholar 

  96. Kröner E (1986) Statistical modeling. In: Gittus J, Zarka J (eds), Modeling Small Deformations of Polycrystals. Elsevier, London/NY, 229–291.

    Chapter  Google Scholar 

  97. Kunin IA (1983) Elastic Media with Microstructure. Springer-Verlag, Berlin, 2

    Google Scholar 

  98. Kurita R, Weeks ER (2010) Experimental study of random-close-packed colloidal particles. Physical Rev., 82:011403

    Google Scholar 

  99. Kuznetsov SV (1991) Microstructural stress in porous media. Priklad Mech, 27(11):23–28 (In Russian. Engl Transl. Soviet Appl Mech, 27:750–755)

    Google Scholar 

  100. Leblond JD, Perrin G (1999) A self-consistent approach to coalescence of cavities in inhomogeneously voided ductile solids. J Mech Phys Solids, 47:1823–1841

    Article  MATH  Google Scholar 

  101. Lee Y, Fang C, Tsou Y-R, Lu L-S, Yang C-T (2009) A packing algorithm for three-dimensional convex particles Granular Matter, 11:307–315

    Google Scholar 

  102. Lingois P, Berglund L (2002) Modeling elastic properties and volume change in dental composites. J. Mater. Sci., 37:4573–4579

    Article  Google Scholar 

  103. Lochmann K, Oger L, Stoyan D (2006) Statistical analysis of random sphere packings with variable radius distribution Solid State Sciences, 8:1397–1413

    Google Scholar 

  104. Lotwick HW (1982) Simulations on some spatial hard core models, and the complete packing problem J Statist Comp Simul, 15:295–314

    Google Scholar 

  105. Lu GQ, Ti LB, Ishizaki K (1994) A new algorithm for simulating the random packing of monosized powder in CIP processes. Mater Manufact Processes, 9:601–621

    Article  Google Scholar 

  106. Lubachevsky BD, Stillinger FH (1990) Geometric properties of random disk packing. J Statist Phys, 60:561–583

    Article  MathSciNet  MATH  Google Scholar 

  107. Lubachevsky BD, Stillinger FH, Pinson EN (1991) Disks vs spheres: contrasting properties of random packing. J Statist Phys, 64:501–524

    Article  MathSciNet  MATH  Google Scholar 

  108. Luijten E (2006) Introduction to cluster Monte Carlo algorithms. Lecture Notes in Physics, 703:13-38

    Article  Google Scholar 

  109. Maekava ZI, Hamada H Yokoyama A (1989) Lamination theory of composite material with complex fiber orientation. Proc ICCS, 5:701–714

    Google Scholar 

  110. Maggi F, Bandera A, Galfetti L, De Luca LT, Jackson TL (2010) Efficient solid rocket propulsion for access to space. Acta Astronautica, 66:1563–1573

    Article  Google Scholar 

  111. Maggi F, Stafford S, Jackson TL, Buckmaster J (2008) Nature of packs used in propellant modeling. Physical Review, E77:046107

    Google Scholar 

  112. Markov KZ, Willis JR (1998) On the two-point correlation function for dispersions of nonoverlap** spheres. Math Models Methods Appl Sci, 8:359–377

    Article  MathSciNet  MATH  Google Scholar 

  113. Massa L, Jackson TL, Short M (2003) Numerical solution of three-dimensional heterogeneous solid propellants. Combust. Theory Modeling, 7:579–602

    Article  Google Scholar 

  114. Matouš K, Geubelle PH (2006) Multiscale modelling of particle debonding in reinforced elastomers subjected to finite deformations. Int. J. Numerical Methods in Engineering, 65:190–223

    Article  MathSciNet  MATH  Google Scholar 

  115. Matouš, K, Inglis H.M, Gu X, Rypl, D, Jackson, T.L, Geubelle, H.P (2007) Multiscale modeling of solid propellants: From particle packing to failure. Composites Science and Technology, 67:1694–1708

    Article  Google Scholar 

  116. Materon G (1989) Random Set and Integral Geometry. John Wiley & Sons, New York

    Google Scholar 

  117. McGeary RK (1961) Mechanical packing of spherical particles. J. Am. Ceram. Soc., 44:513–522

    Article  Google Scholar 

  118. Mishnaevsky LL (2006) Functionally gradient metal matrix composites: Numerical analysis of the microstructure-strength relationships. Comp Sci Technol, 66:1873–1887

    Article  Google Scholar 

  119. Molchanov IS (1997) Statistics of the Boolean Model Model for Practitioners and Mathematicans. John Wiley & Sons, New York

    MATH  Google Scholar 

  120. Monetto I, Drugan WJ (2004) A micromechanics-based nonlocal constitutive equation for elastic composites containing randomly oriented spheroidal heterogeneities J Mech Phys Solids, 52:359–393

    Google Scholar 

  121. Morawiec A, Field DP (1996) Rodrigues parameterization for orientation and misorientation distributions. Philos Mag, A73:1111–1128

    Google Scholar 

  122. Murata I, Mori T, Nakagawa M Continuous energy Monte Carlo calculations of randomly distributed spherical fuels in high-temperature gas-cooled reactor based on a statistical geometry model. Nuclear Sci Engng, 123:96–109

    Google Scholar 

  123. Nigmatulin RI (1979) Spatial averaging in the mechanics of heterogeneous and dispersed systems. Int J Multiphase Flow, 5:353–385

    Article  MATH  Google Scholar 

  124. Nolan GT, Kavanagh PE (1992) Computer simulation of random packing of hard spheres. Powder Technol, 72:149–155

    Article  Google Scholar 

  125. Nolan GT, Kavanagh PE (1995) Octahedral configurations of random close packing. Powder Technology 83:253–258

    Article  Google Scholar 

  126. Nowak ER, Knight JB, Povinelli ML, Jaeger HM, Nagel SR (1997) Reversibility and irreversibility in the packing of vibrated granular material. Powder Technology, 94:79–83

    Article  Google Scholar 

  127. Ogata Y, Tanemura M (1981) Estimation of interaction potentials of spatial point-patterns through the maximum-likelihood procedure. Ann Inst Statist Math, 33:315–338

    Article  MATH  Google Scholar 

  128. Ogata Y, Tanemura M (1984) Likelihood analysis of spatial point-patterns. J Roy Statist Soc, B46:496–518

    MathSciNet  MATH  Google Scholar 

  129. Oger L, Troadec JP, Gervois A, Medvedev N (1998) Computer simulation and tessellations of granular materials. In: Rivier N, Sadoc JF (eds), Foams and Emulsions. Kluver, Dordrecht, 527–545

    Google Scholar 

  130. Ohser J, Mücklich F (2000) Statistical Analysis of Microstructures in Material Science. John Wiley & Sons, Chichester

    MATH  Google Scholar 

  131. Okabe A, Boots B, Sugihara K (1992) Spatial Tessellations. John Wiley & Sons, New York

    MATH  Google Scholar 

  132. Ornstein LS, Zernike F (1914) Accidental deviation of density and opalescence at the critical points of a single substance. Proc Acad Sci (Amsterdam), 17:793–806.

    Google Scholar 

  133. Percus JK, Yevick GJ (1958) Analysis of classical statistical mechanics by means of collective coordinates. Phys Rev, 110:1–13

    Article  MathSciNet  MATH  Google Scholar 

  134. Picka J (2012) Statistical inference for disordered sphere packings. Statist. Surv., 6:74–112

    Article  MathSciNet  MATH  Google Scholar 

  135. Prosperetti A (1998) Ensemble averaging techniques for disperse flows. In: Drew DA, Joseph DD, Passma SL (eds) Particulate Flows Processing and Rheology. Springer-Verlag, New York 99–136

    Chapter  MATH  Google Scholar 

  136. Pyrz R (1994) Quantitative description of the microstructure of composites. Part I: Morphology of unidirectional composite systems. Compos Sci Technol, 50:197–208

    Article  Google Scholar 

  137. Pyrz R (2004) Microstructural description of composites–statistical methods. In: Böhm H (ed), CISM Courses and Lectures. Springer, Udine, 464:173–233

    MATH  Google Scholar 

  138. Pyrz R, Bochenek B (1998) Topological disorder of microstructure and its relation to the stress field. Int J Solids Struct, 35:2413–2427

    Article  MATH  Google Scholar 

  139. Quintanilla J (1999) Microstructure and properties of random heterogeneous materials: a review of theoretical results. Polym Engng Sci, 39:559–585

    Article  Google Scholar 

  140. Quintanilla J, Torquato S (1997) Microstructure functions for a model of statistically inhomogeneous random media. Phys Rev, E55:1558–1565

    Google Scholar 

  141. Rankenburg IC, Zieve RJ (2001) Influence of shape on ordering of granular systems in two dimensions. Phys Rev, E63:61303-1–61303-9

    Google Scholar 

  142. Reiter T, Dvorak GJ, Tvergaard V (1997) Micromechanical models for graded composite materials. J Mech Phys Solids, 45:1281–1302

    Article  Google Scholar 

  143. Ripley BD (1977) Modeling spatial patterns. J Roy Statist Soc, B39:172–212

    Google Scholar 

  144. Ripley BD (1979) Tests of “randomness” for spatial point patterns. J. R. Statist. Soc., B 41:368–374

    Google Scholar 

  145. Ripley BD (1981) Spatial Statistic. John Wiley & Sons, New York

    Book  MATH  Google Scholar 

  146. Rudge JF, Holness MB, Smith GC (2008) Quantitative textural analysis of packings of elongate crystals Contrib Mineral Petrol, 156:413–429

    Google Scholar 

  147. Russ JC (2002) The Image Processing Handbook. CRC Press, Boca Raton, FL

    Book  MATH  Google Scholar 

  148. Russel WB, Saville DA, Schowalter WR (1989) Colloidal Dispersions. Cambridge University Press, Cambridge, UK

    Book  MATH  Google Scholar 

  149. Saadatfar M (2009) Computer simulation of granular materials. Computing in Science & Engineering, 11:66–74

    Article  Google Scholar 

  150. Schaertl W, Sillescu H (1994) Brownian dynamics of polydisperse colloidal hard spheres Equilibrium structures and random close packings. J. Statistical Phys., 77:1007-1025

    Article  Google Scholar 

  151. Segurado J, Llorca J (2002) A numerical approximation to the elastic properties of sphere-reinforced composites. J. Mech. Phys. Solids, 50:2107–2121

    Article  MATH  Google Scholar 

  152. Segurado J, Gonzallez C, Llorca J (2003) A numerical investigation of the effect of particle clustering on the mechanical properties of composites. Acta Mater, 51:2355–2369

    Article  Google Scholar 

  153. Shan Z Gokhale AM (2002) Representative volume element for non-uniform micro-structure. Comput Mater Sci, 24:361–379

    Article  Google Scholar 

  154. Shapiro AP, Probstein RF (1992) Random packings of spheres and fluidity limits of monodisperse and bidisperse suspensions. Phys. Rev. Lett., 68:1422–1425

    Article  Google Scholar 

  155. Shubin AB (1995) On maximum density of random packing of the identical solid spheres. Rasplavy (1):92–97 (In Russian)

    Google Scholar 

  156. Sierou A, Brady JF (2002) Rheology and microstructure in concentrated noncolloidal suspensions. J. Rheology, 46:1031–1056

    Article  Google Scholar 

  157. Silberschmidt V (2008) Account for random microstructure in multiscale models. In Kwon, Y.W, Allen, D.H, Talreja, R.R. (eds) Multiscale Modeling and Simulation of Composite Materials and Structures. Springer, NY. 1–35

    Google Scholar 

  158. Silbert LE, Ertas D, Grest GS, Halsey TC, Levine D (2002) Geometry of frictionless and frictional sphere packings. Physical Review, E65:031304

    MathSciNet  Google Scholar 

  159. Sinelnikov NN, Mazo MA, Berlin AA (1997) Dense packing of random binary assemblies of disks. J Phys I France, 7:247–254

    Article  Google Scholar 

  160. Smith P, Torquato S (1988) Computer simulation results for the two-point probability functions of composite media. J Comput Phys, 76:176–191

    Article  MATH  Google Scholar 

  161. Spitzig WA, Kelly JF, Richmond O (1985) Quantitative characterization of second-phase populations. Metallography, 18:235–261

    Article  Google Scholar 

  162. Spowart JE, Maruyama B, Miracle DB (2001) Multi-scale characterization of spatially heterogeneous systems: implications for discontinuously reinforced metal-matrix composite microstructures. Mater Sci Engng, A307:51–66

    Article  Google Scholar 

  163. Stafford DS, Jackson TL (2010) Using level sets for creating virtual random packs of non-spherical convex shapes. J. Comput. Physics. 229:3295–3315

    Article  MATH  Google Scholar 

  164. Steinhauser OM, Hiermaier S (2009) A review of computational methods in materials science examples from shock-wave and polymer physics. Int. J. Mol. Sci., 10:5135–5216

    Article  Google Scholar 

  165. Stell G (1991) Statistical mechanics applied to random-media problem. In: Owen DRJ, Oñate E, Hinton E (eds) Mathematics of Random Media. Lectures in Applied Mathematics 27:109–127

    Google Scholar 

  166. Stell G, Rirvold PA (1987) Polydispersity in fluid, dispersions, and composites: some theoretical results. Chem Engng Commun, 51:233–260

    Article  Google Scholar 

  167. Stoyan D (1998) Random sets: models and statistics. Int Statistical Rev, 66:1-27

    Article  MATH  Google Scholar 

  168. Stoyan D (2000) Basic ideas of spatial statistics. In: Mecke KR, Stoyan D (eds) Statistical Physics and Spatial Statistics: The Art of Analyzing and Modeling Saptial Structures and Pattern Formation. Lecture Notes in Physics, Springer, Berlin, 554

    Google Scholar 

  169. Stoyan D, Kendall WS, Mecke J (1995) Stochastic Geometry and Its Applications. John Wiley & Sons, Chichester

    MATH  Google Scholar 

  170. Stoyan D, Stoyan H (1994) Fractals, Random Shapes and Point Fields. Methods of Geometric Statistics. J Wiley & Sons, Chichester

    Google Scholar 

  171. Stratonovich RL (1963) Topics in the Theory of Random Noise. Gordon and Breach, New York

    Google Scholar 

  172. Suresh S, Mortensen A (1998) Fundamentals of Functionally Graded Materials : Processing and Thermomechanical Behaviour of Graded Metals and Metal-Ceramic Composites. IOM Communications, London

    Google Scholar 

  173. Tanimoto Y, Nishiwaki T, Nemoto K, Ben G (2004) Effect of filler content on bending properties of dental composites Numerical simulation with the use of the finite-element method. J. Biomedical Materials Research, 71B:188-195

    Article  Google Scholar 

  174. Tanemura M (1979) On random complete packing by discs. Ann Inst Statist Math, 31:351–365

    Article  MATH  Google Scholar 

  175. Taya M (1990) Some thoughts on inhomogeneous distribution of fillers in composites. In: Weng GJ, Taya M, Abe H (eds), Micromechanics and Inhomogeneity, The Toshio Mura 65th Anniversary Volume. Springer-Verlag, New York, 433–447

    Chapter  Google Scholar 

  176. Tewari A, Gokhale AM, Spowart JE, Miracle DB (2004) Quantitative characterization of spatial clustering in three-dimensional microstructures using two-point correlation functions. Acta Mater, 52:307–319

    Article  Google Scholar 

  177. Throop GJ, Bearman RJ (1965) Numerical solution of the Percus-Yervick equation for the hard-sphere potential. J Chem Phys, 42:2408–2411

    Article  MathSciNet  Google Scholar 

  178. Tobochnik J, Chapin PM (1988) Monte Carlo simulation of hard spheres near random closest packing using sphrical boundary conditions. J Chem Phys, 88:5824–5830

    Article  Google Scholar 

  179. Torquato S (2002a) Random Heterogeneous Materials: Microstucture and Macroscopic Properties. Springer-Verlag, New York, Berlin

    Book  MATH  Google Scholar 

  180. Torquato S (2002b) Statistical description of microstructures. Annu Rev Mater Res, 32:77–111

    Article  Google Scholar 

  181. Torquato S, Lado F (1992) Improved bounds on the effective elastic moduli of random arrays of cylinders. J Appl Mech, 59:1–6

    Article  MathSciNet  Google Scholar 

  182. Torquato S, Stell G (1985) Microstructure of two-phase random media. J Chem Phys, 82:980–987

    Article  Google Scholar 

  183. Torquato S, Truskett TM, Debenetti PG (2000) Is random close packing of spheres well defined? Phys Rev Lett, 84:2064-2067

    Article  Google Scholar 

  184. Turnbull D, Cormia RL (1960) A dynamic hard sphere model. J Appl Phys, 31:674–678

    Article  Google Scholar 

  185. Verlet L, Weis JJ (1972) Perturbation theory for the thermodynamic properties of simple liquids. Mol Phys, 24:1013–1024

    Article  Google Scholar 

  186. Webb MD, Davis IL (2006) Random particle packing with large particle size variations using reduced-dimension algorithms. Powder technology, vol. 167, 10-19

    Article  Google Scholar 

  187. Wertheim MS (1963) Exact solution of the Percus-Yevick integral equation for hard spheres. Phys Rev Lett, 10:321–323

    Article  MathSciNet  MATH  Google Scholar 

  188. Widom W (1966) Random sequential addition of hard spheres to a volume. J Chem Phys, 44:3888–3894

    Article  Google Scholar 

  189. Willis JR (1978) Variational principles and bounds for the overall properties of composites. In: Provan JW (ed), Continuum Models of Disordered Systems. University of Waterloo Press, Waterloo 185–215

    Google Scholar 

  190. Wissler M, Lusti HR, Oberson C, Widmann-Schupak AH, Zappini G, Gusev AA (2003) Non-additive effects in the elastic behavior of dental composites. Advanced Engineering Materials, 5:113–116

    Article  Google Scholar 

  191. **a M, Hamada H, Maekawa Z (1995) Flexural stiffness of injection molded glass fiber reinforced thermoplastics. Int Polym Process, 10:74-81

    Article  Google Scholar 

  192. Xu F, Aravas N, Sofronis P (2008) Constitutive modeling of solid propellant materials with evolving microstructural damage. J. Mechanics Physics Solids, 56:2050–2073

    Article  MATH  Google Scholar 

  193. Yang S, Gokhale AM, Shan Z (2000) Utility of microstructure modeling for simulation of micro-mechanical response of composites containing non-uniformly distributed fibers. Acta Mater, 48:2307–2322

    Article  Google Scholar 

  194. Zeman J (2003) Analysis of Composite Materials with Random Microstructures. PhD Thesis. Czech TU in Prague

    Google Scholar 

  195. Ziman JM (1979) Models of Disorder. Cambridge University Press, New York

    Google Scholar 

  196. Zinchenko AZ (1994) Algorithm for random close packing of spheres with periodic boundary conditions. J Comput Phys, 114: 298–307

    Article  MathSciNet  MATH  Google Scholar 

  197. Zou RP, Xu JQ, Feng CL, Yu AB, Johnston S, Standish N (2003) Packing of multi-sized mixtures of wet coarse spheres. Powder Technol., 130:77–83

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Micromechanics and Composites LLC, Cincinnati, OH, USA

    Valeriy A. Buryachenko

Authors
  1. Valeriy A. Buryachenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Cite this chapter

Buryachenko, V.A. (2022). Statistical Description of Composite Materials. In: Local and Nonlocal Micromechanics of Heterogeneous Materials. Springer, Cham. https://doi.org/10.1007/978-3-030-81784-8_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-81784-8_5

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-81783-1

  • Online ISBN: 978-3-030-81784-8

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation