Mechanics of Random Materials

Stochastics, Scale Effects and Computation
  • M. Ostoja-Starzewski
Part of the International Centre for Mechanical Sciences book series (CISM, volume 430)


These notes provide an introduction to three aspects of mechanics of materials with random and multiscale microstructures: stochastic tools, scale effects in constitutive response, and computational methods. The overriding objective in the first Section (Basic Probability Tools) is to give an account of tools of probability theory and random (or so-called stochastic) processes — admittedly a vast subject area per se — at a level sufficient to describe non-deterministic, spatial material fluctuations. An attempt is made here to present random processes as objects parametrized by space coordinates, rather than time. The latter viewpoint is classical throughout most of the literature on random processes, but ours allows a natural generalization to random fields. Indeed, from that standpoint, each and every material microstructure may be considered a random field, although, as pointed out in the chapter by D. Jeulin, a different modeling strategy — based, say, on mathematical morphology — may be more suitable. The presentation covers scalar and vector processes and fields, all illustrated by measurements of paper mechanical properties. This introduces basic concepts of wide-sense stationarity and ergodicity, and sets the base for analyses of random continua.

The second section (Continuum Mechanics of Random Media), central to this chapter, focuses on the problem of Representative Volume Element (RVE) in the sense of Hill (1963). This is, in essence, the problem of coupled size and boundary condition dependence of constitutive response of microstructures described by random fields. Indeed, given three possible types of boundary conditions, we have three possible apparent responses, two of which (kinematic- and traction-controlled) hound on mesoscale the effective (in the macroscopic sense) response. The mesoscale domain involved is called a Statistical Volume Element (SVE). A review is provided here of the results obtained in this field over the past dozen years. We first study hierarchies of mesoscale bounds for elastic microstructures: both qualitative results via variational principles and quantitative results-via computational mechanics. The same type of approach is then sketched for inelastic microstructures, although a much wider range of results is given in the elastic case. This chapter culminates in a short review of two topics in wave propagation in random media: spectral finite elements and wavefront propagation. Here the wave length, respectively wavefront thickness, again introduces an SVE in place of an RVE.

The third section (Some Computational Mechanics Methods for Random Media) directly builds on, and supports, the second one. On one hand, it is shown how the mesoscale bounds can be used as input into the finite element methods, thus leading to micromechanics-based stochastic finite elements. The semblance, as well as contradistinction, of this approach to conventional finite elements and conventional random field models of continua is discussed. On the other hand; we provide a very brief introduction into spring network models, heavily favored by this author for rapid computation of statistics of mesoscale bounds of elastic materials, based on Monte Carlo-type generation of their random microstructures.


Random Field Displacement Boundary Condition Hill Condition Representative Volume Element Size Random Material 
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.


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  1. Alzebdeh, K., Al-Ostaz, A., Jasiuk, I. and Ostoja-Starzewski, M. (1998), Fracture of random matrix-inclusion composites: scale effects and statistics, Intl. J. Solids Struct. 35 (19), 2537–2566.CrossRefMATHGoogle Scholar
  2. Alzebdeh, K. and Ostoja-Starzewski, M. (1999), On a spring network model and effective elastic moduli of granular materials, ASME J. Appl. Mech. 66, 172–180.CrossRefGoogle Scholar
  3. Ashby, M.F. and Jones, D.R.H. (1986), Engineering Materials 2, Pergamon Press, Oxford.Google Scholar
  4. Askar, A. (1985). Lattice Dynamical Foundations of Continuum Theories, World Scientific, Singapore.Google Scholar
  5. Bazant, Z.P. and Planas, J. (1998), Fracture and Size Effect in Concrete and Other Quasibrittle Materials, CRC Press, Boca Raton.Google Scholar
  6. Beale; P.D. and Srolowitz, D.J. (1988), Elastic fracture in random materials, Phys. Rev. 337(10), 5500–5507.Google Scholar
  7. Belyaev, A.K. and Ziegler, F. (1994), Effective loss factor of heterogeneous elastic solids and fluids, in Trends in Applications of Mathematics to Mechanics (M.D.P. Monteiro Marques and J.F. Rodrigues, eds. ), 3–13.Google Scholar
  8. Benaroya, H. and Rehak, M. (1988), Finite element methods in probabilistic structural analysis: a selective review, Appl. Mech. Rev. 41 (5), 201–213.CrossRefGoogle Scholar
  9. Beran. M.J. (1968), Statistical Continuum Theories, Interscience, New York.Google Scholar
  10. Beran, M.J. (1974). Application of statistical theories for the determination of thermal, electrical, and magnetic properties of heterogeneous materials, in Mechanics of Composite Materials (ed. G.P. Sendeckyj) 2, 209–249.Google Scholar
  11. Beran, M.J. (2001), this volume.Google Scholar
  12. Brenner, C.E. (1991), Stochastic Finite Elements (Literature Review), Internal Working Report 35–91, Institute of Engineering Mechanics, University of Innsbruck.Google Scholar
  13. Brezzi, F. and Fortin, M. (1991), Mixed and Hybrid Finite Element Methods, Springer-Verlag, Berlin.CrossRefMATHGoogle Scholar
  14. Budiansky, B. and O’Connell, R.J. (1976), Elastic moduli of a cracked solid, Intl. J. Solids Struct. 12, 81–97.CrossRefMATHGoogle Scholar
  15. Carvalho, F.C.S. and Labuz, J.F. (1996), Experiments on effective elastic modulus of two-dimensional solids with cracks and holes, Intl. J. Solids Struct. 33 (28), 4119–4130.CrossRefGoogle Scholar
  16. Cash, J.R. and Wright, M.H. (1991), A deferred correction method for nonlinear two-point boundary value problems: Implementation and numerical evaluation, SIAM J. Sei. Stat. Comput. 12, 971–989.CrossRefMATHMathSciNetGoogle Scholar
  17. Chen, J.Y., Huang, Y. and Ortiz, M. (1998), Fracture analysis of cellular materials, J. Mech. Phys. Solids 46, 789–828.CrossRefMATHMathSciNetGoogle Scholar
  18. Christensen, R.M. (1982), Theory of Viscoelasticity: an Introduction, Academic Press, New York.Google Scholar
  19. Cioranescu, D. and Saint Jean Paulin, J.. (1999), Homogenization of Reticulated Structures, Springer Verlag, New York.CrossRefMATHGoogle Scholar
  20. Coleman, B.D. and Curtin, M.E. (1965), Arch. Rational Mech. Anal. 19, 239–265.CrossRefMATHMathSciNetGoogle Scholar
  21. Contreras, H. (1980), The stochastic finite-element method, Computers and Structures 12, 341–348.CrossRefMATHMathSciNetGoogle Scholar
  22. Costello, G.A. (1997), Theory of Wire Rope, Springer-Verlag, Berlin.CrossRefGoogle Scholar
  23. Dempsey, J.P. (2000), Research trends in ice mechanics, in ( G.J. Dvorak, ed.), Research Trends in Solid Mechanics, Intl. J. Solids Struct. 37(1–2), 131–153.Google Scholar
  24. Deodatis, G., Wall, W. and Shinozuka, M. (1991), Analysis of two-dimensional stochastic systems by the weighted integral method, in Computational Stochastic Mechanics (P.D. Spanos and C.A. Brebbia, eds., CMP-Elsevier, 395–406.Google Scholar
  25. DiMillo, Ivi. and Ostoja-Starzewski, M. (1998), Paper strength: Statistics and correlation structure, Intl. J. Fract. 90, L33 - L38.Google Scholar
  26. Ditlevsen, O. (1996). Dimension reduction and discretization in stochastic problems by regression method, in Mathematical Models for Structural Reliability Analysis, 51–138, CRC Press, Boca Raton, FL.Google Scholar
  27. Doyle, J.F. (1997), Wave Propagation in Structures: Spectral Analysis Using Fast Discrete Fourier Transforms, Springer-Verlag, Berlin.CrossRefMATHGoogle Scholar
  28. Drugan, W.J. and Willis, J.R. (1996), A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites, J. 11MIec..h. Phys. Solids 44, 497–524.CrossRefMATHMathSciNetGoogle Scholar
  29. Frisch, U. (1968), Wave propagation in random media, in Probabilistic Methods in Applied Mathematics 1 (A.T. Bharucha-Reid, ed.), 75–198, Academic Press.Google Scholar
  30. Ghanem, R. and Spanos, P.D. ( 1991. ), Stochastic Finite Elements: a Spectral Approach, Springer-Verlag, Berlin.Google Scholar
  31. Gusev, A.A. (1997), Representative volume element size for elastic composites: A numerical study, J. Mech. Phys. Solids 45, 1449.CrossRefMATHGoogle Scholar
  32. Hazanov, S. (1998), Hill condition and overall properties of composites, Arch. Appl. Mech. 66, 385–394.CrossRefGoogle Scholar
  33. Hazanov, S. (1999). On apparent properties of nonlinear heterogeneous bodies smaller than the representative volume, Acta Mech. 134, 123–134.CrossRefMATHMathSciNetGoogle Scholar
  34. Hazanov. S. and Amieur, M. (1995), On overall properties of elastic heterogeneous bodies smaller than the representative volume, Intl. J. Eng. Sci. 33 (9), 1289–1301.CrossRefGoogle Scholar
  35. Hazanov, S. and Huet, C. (1994), Order relationships for boundary conditions effect in heterogeneous bodies smaller than the representative volume, J. Mech. Phys. Solids 41, 1995–2011.CrossRefMathSciNetGoogle Scholar
  36. Herrmann, H. J., and Roux, S., eds., (1990), Statistical Models for Fracture of Disordered Media,Elsevier.Google Scholar
  37. Hill, R. (1950), The Mathematical Theory of Plasticity, Oxford University Press, London.MATHGoogle Scholar
  38. Hill, R. (1963), Elastic properties of reinforced solids: Some theoretical principles, J. Mech. Phys. Solids 11, 357–372.CrossRefMATHGoogle Scholar
  39. Huet. C. (1982), Universal conditions for assimilation of a heterogeneous mate. rial to an effective medium, Mech. Res. Comm. 9 (3), 165–170.CrossRefMATHGoogle Scholar
  40. Huet, C. (1990), Application of variational concepts to size effects in elastic heterogeneous bodies, J. Mech. Phys. Solids 38, 813–841.CrossRefMathSciNetGoogle Scholar
  41. Huet, C. (1991), Hierarchies and bounds for size effects in heterogeneous bodies, Continuum Models and Discrete Systems 2 (G.A. Maugin, ed.), 127–134, Longman Scientific and Technical.Google Scholar
  42. Huet, C. (1994), Experimental characterization, micromechanical simulation and spatio-stochastic approach of concrete behaviours below the representative volume, PROBAMAT - Probabilities and Materials: Tests, Models and Applications (D. Breysse, ed.), Proc. NATO Adv. Res. Workshop Series E: 269, 241–260, Kluwer.Google Scholar
  43. Huet, C. (1995), Bounds and hierarchies for the overall properties of viscoelastic heterogeneous and composite materials, Arch. Mech. 47 (6), 1125–1155.MATHMathSciNetGoogle Scholar
  44. Huet, C. (1999a), Strength scaling law for elastic materials with interacting defects, in Mechanics of Quasi-brittle Materials and Structures ( G. Pijaudier-Cabot, Z. Bittnar and B. Gérard, eds.), 31–38, Hermes, Paris.Google Scholar
  45. Huet, C. (1999b), Coupled size and boundary-condition effects in viscoelastic heterogeneous and composite bodies, Mech. Mater. 31 (12), 787–829.CrossRefGoogle Scholar
  46. Huet, C. (2001), Extended Clapeyron formulae for viscoelasticity problems in the time domain and application to the boundary condition effect in random composite bodies, J. Mech. Phys. Solids 49, 675–706.CrossRefMATHMathSciNetGoogle Scholar
  47. Jeulin, D. (1997), Advances in Theory and Applications of Random Sets, World Scientific, Singapore.MATHGoogle Scholar
  48. Jeulin, D. (2001), this volume.Google Scholar
  49. Jiang, M., Alzebdeh, K., Jasiuk, L and Ostoja-Starzewski, M. (2001a), Scale and boundary conditions effects in elasticity of random composites, Acta Mech. 148 (1–4), 63–78.CrossRefMATHGoogle Scholar
  50. Jiang, M., Ostoja-Starzewski, M. and Jasiuk, I. (2001b), Scale-dependent bounds on effective elastoplastic response of random composites, J. Mech. Phys. Solids 49 (3), 655–673.CrossRefMATHGoogle Scholar
  51. Kachanov, L.M. (1971), Foundations of the Theory of Plasticity, North-Holland, Amsterdam.MATHGoogle Scholar
  52. Kachanov, M. (1993), Elastic solids with many cracks and related problems, Adv. Appl. Mech. 30, 259–445.CrossRefGoogle Scholar
  53. Krajcinovic, D. (1996), Damage Mechanics, North-Holland, Amsterdam.Google Scholar
  54. Kröner, E. (1972), Statistical Continuum Mechanics, CISM Lectures 92, Springer-Verlag, Wien.Google Scholar
  55. Kröner, E. (1994), Nonlinear elastic properties of microheterogeneous media, ASME J. Eng. Mater. Tech. 116, 325–330.CrossRefGoogle Scholar
  56. Kruyt, N.P. and Rothenburg, L. (2001), Statics of the elastic behaviour of granular materials, Int. J. Solids Struct. 38, 4879–4889.CrossRefMATHGoogle Scholar
  57. Lemaitre, J. and Chaboche, J.-L. (1994), Mechanics of Solid Materials,Cambridge University Press.Google Scholar
  58. Liu, W.K., Belytschko, T. and Lua, Y.J. (1995), Probabilistic finite element method, in Probabilistic Structural Mechanics Handbook (C. Sundararajan, ed.), 70–105, Chapman and Hall, New York.Google Scholar
  59. Matheron, G. (1974), Random Sets and Integral Geometry,J. Wiley and Sons.Google Scholar
  60. Monette, L. and Anderson, M.P. (1999), Application of lattice models to fracture of heterogeneous materials under biaxial loads, Applied Mechanics in the Americas - 6th Pan American Congress of Applied Mechanics 7 ( Eds. P.B. Gonçalves, I. JasiukGoogle Scholar
  61. D. Pamplona, C. Steele, I. Weber and L. Bevilacqua), Rio de Janeiro, Brazil, 369–372.Google Scholar
  62. Noor, A.K. (1988), Continuum modeling for repetitive lattice structures, Appl. Mech. Rev. 41 (7), 285–296.CrossRefGoogle Scholar
  63. Ostoja-Starzewski, M. (1993), Micromechanics as a basis of stochastic finite elements and differences - an overview, Appl. Mech. Rev. (Special Issue: Mechanics Pan-America 1993 ) 46 (11, Pt. 2), S136 - S147.Google Scholar
  64. Ostoja-Starzewski, M. (1994), Micromechanics as a basis of continuum random fields, Appl. Mech. Rev. (Special Issue: Micromechanics of Random Media) 47 (1, Pt. 2), S221 - S230.Google Scholar
  65. Ostoja-Starzewski, M. (1998), Random field models of heterogeneous materials, Intl. J. Solids Struct. 35 (19), 2429–2455.CrossRefMATHGoogle Scholar
  66. Ostoja-Starzewski, M. (1999a), Scale effects in materials with random distributions of needles and cracks, Mech. Mater. 31 (12), 883–893.CrossRefGoogle Scholar
  67. Ostoja-Starzewski, M. (1999b), Microstructural disorder, mesoscale finite elements, and macroscopic response, Proc. R. Soc. Lond. A455, 3189–3199.CrossRefMATHMathSciNetGoogle Scholar
  68. Ost.oja-Starzewski, M. (2000), Universal material property in conductivity of planar random microstructures, Phys. Rev. B62 (5), 2980–2983.CrossRefGoogle Scholar
  69. Ostoja-Starzewski, M. (2001), Microstructural randomness versus representative volume element in thermomechanics, ASME J. Appl. Mech.,in press.Google Scholar
  70. Ostoja-Starzewski, M. (2001/02), Lattice models in rnicromechanics, Appl. Mech. Rev.,in press.Google Scholar
  71. Ostoja-Starzewski, M., Jasiuk, I., Wang, W. and Alzebdeh, K. (1996), Composites with functionally graded interfaces: Meso-continuum concept and effective properties, Acta Mater. 44 (5), 2057–2066.CrossRefGoogle Scholar
  72. Ostoja-Starzewski, M., Quadrelli, M.B., and Stahl, D.C. (1999), Kinematics and stress transfer in quasi-planar random fiber networks, C. R. Acad. Sci. Paris IIG 327, 1223–1229.MATHGoogle Scholar
  73. Ostoja-Starzewski, M. and Schulte, J. (1996), Bounding of effective thermal conductivities of rnultiscale materials by essential and natural boundary conditions, Phys. Rev. B54 (1), 278–285.CrossRefGoogle Scholar
  74. Ostoja-Starzewski, M. and J. Trebicki, J. (1999), On the growth and decay of acceleration waves in random media, Proc. R. Soc. Lond. A455, 2577–2614.CrossRefMATHMathSciNetGoogle Scholar
  75. Ostoja-Starzewski, M. and Wang, C. (1989), Linear elasticity of planar Delaunay networks: Random field characterization of effective moduli, Acta Mech. 80, 61–80.CrossRefMATHMathSciNetGoogle Scholar
  76. Ostoja-Starzewski, M. and Wang, X. (1999), Stochastic finite elements as a bridge between random material microstructure and global response, Comp. Meth. Appl. Mech. Eng. (Special Issue: Computational Stochastic Mechanics), 168 (1–4), 35–49.MATHGoogle Scholar
  77. Ponte Castaneda, P. and Suquet, P. (1998), Nonlinear composites, Adv. Appl. Mech. 34, 171–302.CrossRefGoogle Scholar
  78. Prohorov, Yu.V. and Rozanov, Yu.A. (1969), Probability Theory, Springer-Verlag, Berlin.CrossRefGoogle Scholar
  79. Pshenichnov, G. I. (1993), A Theory of Latticed Plates and Shells, World Scientific, Singapore.CrossRefMATHGoogle Scholar
  80. Reuss, A. (1929), Berechnung der Fliessgrenze Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle, ZAMM 9, 49–58.CrossRefMATHGoogle Scholar
  81. Rytov, S.M., Kravtsov, Yu.A., Tatarskii, V.I. (1987), Principles of Statistical Radiophysics, Springer-Verlag, Berlin.CrossRefMATHGoogle Scholar
  82. Sah. K. (1991), Principe de Hill et homogénéisation des matériaux aléatoires, C. R. Acad. Sci. Paris II 312, 1–5.Google Scholar
  83. Sab, K. (1992), On the homogenization and the simulation of random materials, Europ. J. Mech., A/Solids 11, 585–607.MATHMathSciNetGoogle Scholar
  84. Sab, K. (1994), Homogenization of nonlinear random media by a duality method. Application to plasticity, Asymptotic Anal. 9, 311–336.MATHMathSciNetGoogle Scholar
  85. Sachs, G. (1928), Zur Ableitung einer Fliessbedingung, Z. Ver. Deutsch. Ing. 72, 734–736.Google Scholar
  86. Saigal, S. (1995), Stochastic boundary elements for two-dimensional potential flow in non-homogeneous media, Comp. Meth. Appl. Mech. Eng. 121, 211–230.CrossRefMATHGoogle Scholar
  87. Sobczyk, K. (1985), Stochastic Wave Propagation, Elsevier-Polish Sci. Publ., Amsterdam-Warsaw.Google Scholar
  88. Stolz, C. (1986), General relationships between micro and macro scales for the non-linear behaviour of heterogeneous media, Modelling Small Deformations of Polycrystals (J. Gittus and J. Zarka, eds. ), 89–115.Google Scholar
  89. Stoyan, D., Kendall, W.S. and Mecke, J. (1987), Stochastic Geometry and its Applications,J. Wiley and Sons, New York.Google Scholar
  90. Suquet, P.M. (1986), Elements of homogenization for solid mechanics, in Homogenization Techniques for Composite Media, Lecture Notes in Physics 272, (ed. E. Sanchez-Palencia. and A. Zaoui ), 193–278.Google Scholar
  91. Suquet, P. ed. (1997), Continuum Micromechanics, Springer-Verlag, Wien.MATHGoogle Scholar
  92. Taylor, G.I. (1938), Plastic strain in metals, J. Inst. Met. 62, 307–324.Google Scholar
  93. Terada, K., Hori, M., Kyoya, T. and Kikuchi, N. (2000), Simulation of the multi-scale convergence in computational homogenization approaches, Int. J. Solids Struct. 37, 2285–2311.CrossRefMATHGoogle Scholar
  94. Torquato, S. (1991), Random heterogeneous media: microstructure and improved bounds on effective properties, Appl. Mech. Rev. 44, 37–76.CrossRefMathSciNetGoogle Scholar
  95. Vanmarcke, E.H. (1983), Random Fields: Analysis and Synthesis, MIT Press, Cambridge, MA.Google Scholar
  96. Vakulenko, A.A. and Kachanov, M.L. (1971), Continuum model of a medium with cracks (in Russian), Mech. Solids 4, 54–59.Google Scholar
  97. Van Mier, J.G.M. (1997), Fracture Processes of Concrete, CRC Press, Boca Raton, FL.Google Scholar
  98. Voigt; W. (1889), Über die Beziehung zwischen den beiden Elastizitätskonstanten isotroper Körper, Wied. Ann. 38 573–587.Google Scholar
  99. Willis, J.R. (1981), Variational and related methods for the overall properties of composites, Adv. Appl. Mech. 21, 1–78.CrossRefMATHMathSciNetGoogle Scholar
  100. Willis, J.R. and Talbot, D.R.S. (1989), in Prue . 6th Symp. Continuum Models and Discrete Systems (ed. G.A. Maugin ), Longmans, London.Google Scholar
  101. Willis, J.R. (2001), this volume.Google Scholar
  102. Wozniak, C. (1970), Surface Lattice Structures (in Polish), Polish Sci. Publ., Warsaw.Google Scholar
  103. Yaglom, A.M. (1957), An Introduction to the Theory of Stationary Random Functions, Dover, New York.Google Scholar
  104. Zubelewicz, A. and Bazant, Z.P. (1987), Interface modeling of fracture in aggregate composites, ASCE J. Eng. Mech. 113, 1619–1630.CrossRefGoogle Scholar

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© Springer-Verlag Wien 2001

Authors and Affiliations

  • M. Ostoja-Starzewski
    • 1
  1. 1.McGill UniversityMontrèalCanada

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