# Mechanics of Random Materials

## Summary

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.

## Keywords

Random Field Displacement Boundary Condition Hill Condition Representative Volume Element Size Random Material## Preview

Unable to display preview. Download preview PDF.

## References

- 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 - 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 - Ashby, M.F. and Jones, D.R.H. (1986),
*Engineering Materials*2, Pergamon Press, Oxford.Google Scholar - Askar, A. (1985).
*Lattice Dynamical Foundations of Continuum Theories*, World Scientific, Singapore.Google Scholar - Bazant, Z.P. and Planas, J. (1998),
*Fracture and Size Effect in Concrete and Other Quasibrittle Materials*, CRC Press, Boca Raton.Google Scholar - Beale
_{;}P.D. and Srolowitz, D.J. (1988), Elastic fracture in random materials,*Phys. Rev.***337**(10), 5500–5507.Google Scholar - 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 - 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 - Beran. M.J. (1968),
*Statistical Continuum Theories*, Interscience, New York.Google Scholar - 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 - Beran, M.J. (2001), this volume.Google Scholar
- Brenner, C.E. (1991),
*Stochastic Finite Elements*(Literature Review), Internal Working Report 35–91, Institute of Engineering Mechanics, University of Innsbruck.Google Scholar - Brezzi, F. and Fortin, M. (1991),
*Mixed and Hybrid Finite Element Methods*, Springer-Verlag, Berlin.CrossRefMATHGoogle Scholar - Budiansky, B. and O’Connell, R.J. (1976), Elastic moduli of a cracked solid,
*Intl. J. Solids Struct.***12**, 81–97.CrossRefMATHGoogle Scholar - 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 - 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 - Chen, J.Y., Huang, Y. and Ortiz, M. (1998), Fracture analysis of cellular materials,
*J. Mech. Phys. Solids***46**, 789–828.CrossRefMATHMathSciNetGoogle Scholar - Christensen, R.M. (1982),
*Theory of Viscoelasticity: an Introduction*, Academic Press, New York.Google Scholar - Cioranescu, D. and Saint Jean Paulin, J.. (1999),
*Homogenization of Reticulated Structures*, Springer Verlag, New York.CrossRefMATHGoogle Scholar - Coleman, B.D. and Curtin, M.E. (1965),
*Arch. Rational Mech. Anal.***19**, 239–265.CrossRefMATHMathSciNetGoogle Scholar - Contreras, H. (1980), The stochastic finite-element method,
*Computers and Structures***12**, 341–348.CrossRefMATHMathSciNetGoogle Scholar - Costello, G.A. (1997),
*Theory of Wire Rope*, Springer-Verlag, Berlin.CrossRefGoogle Scholar - 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 - 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 - DiMillo, Ivi. and Ostoja-Starzewski, M. (1998), Paper strength: Statistics and correlation structure,
*Intl. J. Fract.***90**, L33 - L38.Google Scholar - 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 - Doyle, J.F. (1997),
*Wave Propagation in Structures: Spectral Analysis Using Fast Discrete Fourier Transforms*, Springer-Verlag, Berlin.CrossRefMATHGoogle Scholar - 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 - 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 - Ghanem, R. and Spanos, P.D. ( 1991. ),
*Stochastic Finite Elements: a Spectral Approach*, Springer-Verlag, Berlin.Google Scholar - Gusev, A.A. (1997), Representative volume element size for elastic composites: A numerical study,
*J. Mech. Phys. Solids***45**, 1449.CrossRefMATHGoogle Scholar - Hazanov, S. (1998), Hill condition and overall properties of composites,
*Arch. Appl. Mech.***66**, 385–394.CrossRefGoogle Scholar - Hazanov, S. (1999). On apparent properties of nonlinear heterogeneous bodies smaller than the representative volume,
*Acta Mech*.**134**, 123–134.CrossRefMATHMathSciNetGoogle Scholar - 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 - 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 - Herrmann, H. J., and Roux, S., eds., (1990),
*Statistical Models for Fracture of Disordered Media*,Elsevier.Google Scholar - Hill, R. (1950),
*The Mathematical Theory of Plasticity*, Oxford University Press, London.MATHGoogle Scholar - Hill, R. (1963), Elastic properties of reinforced solids: Some theoretical principles,
*J. Mech. Phys. Solids***11**, 357–372.CrossRefMATHGoogle Scholar - 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 - Huet, C. (1990), Application of variational concepts to size effects in elastic heterogeneous bodies,
*J. Mech. Phys. Solids***38**, 813–841.CrossRefMathSciNetGoogle Scholar - 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 - 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 - Huet, C. (1995), Bounds and hierarchies for the overall properties of viscoelastic heterogeneous and composite materials,
*Arch. Mech.***47**(6), 1125–1155.MATHMathSciNetGoogle Scholar - 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 - Huet, C. (1999b), Coupled size and boundary-condition effects in viscoelastic heterogeneous and composite bodies,
*Mech. Mater.***31**(12), 787–829.CrossRefGoogle Scholar - 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 - Jeulin, D. (1997),
*Advances in Theory and Applications of Random Sets*, World Scientific, Singapore.MATHGoogle Scholar - Jeulin, D. (2001), this volume.Google Scholar
- 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 - 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 - Kachanov, L.M. (1971),
*Foundations of the Theory of Plasticity*, North-Holland, Amsterdam.MATHGoogle Scholar - Kachanov, M. (1993), Elastic solids with many cracks and related problems,
*Adv. Appl. Mech.***30**, 259–445.CrossRefGoogle Scholar - Krajcinovic, D. (1996),
*Damage Mechanics*, North-Holland, Amsterdam.Google Scholar - Kröner, E. (1972),
*Statistical Continuum Mechanics*, CISM Lectures**92**, Springer-Verlag, Wien.Google Scholar - Kröner, E. (1994), Nonlinear elastic properties of microheterogeneous media,
*ASME J. Eng. Mater. Tech.***116**, 325–330.CrossRefGoogle Scholar - Kruyt, N.P. and Rothenburg, L. (2001), Statics of the elastic behaviour of granular materials,
*Int. J. Solids Struct.***38**, 4879–4889.CrossRefMATHGoogle Scholar - Lemaitre, J. and Chaboche, J.-L. (1994),
*Mechanics of Solid Materials*,Cambridge University Press.Google Scholar - 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 - Matheron, G. (1974),
*Random Sets and Integral Geometry*,J. Wiley and Sons.Google Scholar - 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 - D. Pamplona, C. Steele, I. Weber and L. Bevilacqua), Rio de Janeiro, Brazil, 369–372.Google Scholar
- Noor, A.K. (1988), Continuum modeling for repetitive lattice structures, Appl.
*Mech. Rev.***41**(7), 285–296.CrossRefGoogle Scholar - 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 - 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 - Ostoja-Starzewski, M. (1998), Random field models of heterogeneous materials,
*Intl. J. Solids Struct.***35**(19), 2429–2455.CrossRefMATHGoogle Scholar - Ostoja-Starzewski, M. (1999a), Scale effects in materials with random distributions of needles and cracks,
*Mech. Mater.*31 (12), 883–893.CrossRefGoogle Scholar - Ostoja-Starzewski, M. (1999b), Microstructural disorder, mesoscale finite elements, and macroscopic response,
*Proc. R. Soc. Lond.*A455, 3189–3199.CrossRefMATHMathSciNetGoogle Scholar - Ost.oja-Starzewski, M. (2000), Universal material property in conductivity of planar random microstructures,
*Phys*. Rev.**B62**(5), 2980–2983.CrossRefGoogle Scholar - Ostoja-Starzewski, M. (2001), Microstructural randomness versus representative volume element in thermomechanics,
*ASME J. Appl. Mech.*,in press.Google Scholar - Ostoja-Starzewski, M. (2001/02), Lattice models in rnicromechanics, Appl.
*Mech. Rev.*,in press.Google Scholar - 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 - 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 - 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 - 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 - 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 - 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 - Ponte Castaneda, P. and Suquet, P. (1998), Nonlinear composites,
*Adv. Appl. Mech.***34**, 171–302.CrossRefGoogle Scholar - Prohorov, Yu.V. and Rozanov, Yu.A. (1969),
*Probability Theory*, Springer-Verlag, Berlin.CrossRefGoogle Scholar - Pshenichnov, G. I. (1993),
*A Theory of Latticed Plates and Shells*, World Scientific, Singapore.CrossRefMATHGoogle Scholar - Reuss, A. (1929), Berechnung der Fliessgrenze Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle,
*ZAMM***9**, 49–58.CrossRefMATHGoogle Scholar - Rytov, S.M., Kravtsov, Yu.A., Tatarskii, V.I. (1987),
*Principles of Statistical Radiophysics*, Springer-Verlag, Berlin.CrossRefMATHGoogle Scholar - Sah. K. (1991), Principe de Hill et homogénéisation des matériaux aléatoires,
*C. R. Acad. Sci. Paris II***312**, 1–5.Google Scholar - Sab, K. (1992), On the homogenization and the simulation of random materials,
*Europ. J. Mech.*,*A/Solids***11**, 585–607.MATHMathSciNetGoogle Scholar - Sab, K. (1994), Homogenization of nonlinear random media by a duality method. Application to plasticity,
*Asymptotic Anal*.**9**, 311–336.MATHMathSciNetGoogle Scholar - Sachs, G. (1928), Zur Ableitung einer Fliessbedingung,
*Z. Ver. Deutsch. Ing.***72**, 734–736.Google Scholar - 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 - Sobczyk, K. (1985),
*Stochastic Wave Propagation*, Elsevier-Polish Sci. Publ., Amsterdam-Warsaw.Google Scholar - 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 - Stoyan, D., Kendall, W.S. and Mecke, J. (1987),
*Stochastic Geometry and its Applications*,J. Wiley*and*Sons, New York.Google Scholar - 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 - Suquet, P. ed. (1997),
*Continuum Micromechanics*, Springer-Verlag, Wien.MATHGoogle Scholar - 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 - Torquato, S. (1991), Random heterogeneous media: microstructure and improved bounds on effective properties,
*Appl. Mech. Rev.***44**, 37–76.CrossRefMathSciNetGoogle Scholar - Vanmarcke, E.H. (1983),
*Random Fields: Analysis and Synthesis*, MIT Press, Cambridge, MA.Google Scholar - Vakulenko, A.A. and Kachanov, M.L. (1971), Continuum model of a medium with cracks (in Russian),
*Mech. Solids***4**, 54–59.Google Scholar - Van Mier, J.G.M. (1997),
*Fracture Processes of Concrete*, CRC Press, Boca Raton, FL.Google Scholar - Voigt
_{;}W. (1889), Über die Beziehung zwischen den beiden Elastizitätskonstanten isotroper Körper,*Wied. Ann.***38**573–587.Google Scholar - Willis, J.R. (1981), Variational and related methods for the overall properties of composites,
*Adv. Appl. Mech.***21**, 1–78.CrossRefMATHMathSciNetGoogle Scholar - 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 - Willis, J.R. (2001), this volume.Google Scholar
- Wozniak, C. (1970),
*Surface Lattice Structures*(in Polish), Polish Sci. Publ., Warsaw.Google Scholar - Yaglom, A.M. (1957),
*An Introduction to the Theory of Stationary Random Functions*, Dover, New York.Google Scholar - Zubelewicz, A. and Bazant, Z.P. (1987), Interface modeling of fracture in aggregate composites,
*ASCE J. Eng. Mech.***113**, 1619–1630.CrossRefGoogle Scholar