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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
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.
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.
Ashby, M.F. and Jones, D.R.H. (1986), Engineering Materials 2, Pergamon Press, Oxford.
Askar, A. (1985). Lattice Dynamical Foundations of Continuum Theories, World Scientific, Singapore.
Bazant, Z.P. and Planas, J. (1998), Fracture and Size Effect in Concrete and Other Quasibrittle Materials, CRC Press, Boca Raton.
Beale; P.D. and Srolowitz, D.J. (1988), Elastic fracture in random materials, Phys. Rev. 337(10), 5500–5507.
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.
Benaroya, H. and Rehak, M. (1988), Finite element methods in probabilistic structural analysis: a selective review, Appl. Mech. Rev. 41 (5), 201–213.
Beran. M.J. (1968), Statistical Continuum Theories, Interscience, New York.
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.
Beran, M.J. (2001), this volume.
Brenner, C.E. (1991), Stochastic Finite Elements (Literature Review), Internal Working Report 35–91, Institute of Engineering Mechanics, University of Innsbruck.
Brezzi, F. and Fortin, M. (1991), Mixed and Hybrid Finite Element Methods, Springer-Verlag, Berlin.
Budiansky, B. and O’Connell, R.J. (1976), Elastic moduli of a cracked solid, Intl. J. Solids Struct. 12, 81–97.
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.
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.
Chen, J.Y., Huang, Y. and Ortiz, M. (1998), Fracture analysis of cellular materials, J. Mech. Phys. Solids 46, 789–828.
Christensen, R.M. (1982), Theory of Viscoelasticity: an Introduction, Academic Press, New York.
Cioranescu, D. and Saint Jean Paulin, J.. (1999), Homogenization of Reticulated Structures, Springer Verlag, New York.
Coleman, B.D. and Curtin, M.E. (1965), Arch. Rational Mech. Anal. 19, 239–265.
Contreras, H. (1980), The stochastic finite-element method, Computers and Structures 12, 341–348.
Costello, G.A. (1997), Theory of Wire Rope, Springer-Verlag, Berlin.
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.
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.
DiMillo, Ivi. and Ostoja-Starzewski, M. (1998), Paper strength: Statistics and correlation structure, Intl. J. Fract. 90, L33 - L38.
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.
Doyle, J.F. (1997), Wave Propagation in Structures: Spectral Analysis Using Fast Discrete Fourier Transforms, Springer-Verlag, Berlin.
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.
Frisch, U. (1968), Wave propagation in random media, in Probabilistic Methods in Applied Mathematics 1 (A.T. Bharucha-Reid, ed.), 75–198, Academic Press.
Ghanem, R. and Spanos, P.D. ( 1991. ), Stochastic Finite Elements: a Spectral Approach, Springer-Verlag, Berlin.
Gusev, A.A. (1997), Representative volume element size for elastic composites: A numerical study, J. Mech. Phys. Solids 45, 1449.
Hazanov, S. (1998), Hill condition and overall properties of composites, Arch. Appl. Mech. 66, 385–394.
Hazanov, S. (1999). On apparent properties of nonlinear heterogeneous bodies smaller than the representative volume, Acta Mech. 134, 123–134.
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.
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.
Herrmann, H. J., and Roux, S., eds., (1990), Statistical Models for Fracture of Disordered Media,Elsevier.
Hill, R. (1950), The Mathematical Theory of Plasticity, Oxford University Press, London.
Hill, R. (1963), Elastic properties of reinforced solids: Some theoretical principles, J. Mech. Phys. Solids 11, 357–372.
Huet. C. (1982), Universal conditions for assimilation of a heterogeneous mate. rial to an effective medium, Mech. Res. Comm. 9 (3), 165–170.
Huet, C. (1990), Application of variational concepts to size effects in elastic heterogeneous bodies, J. Mech. Phys. Solids 38, 813–841.
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.
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.
Huet, C. (1995), Bounds and hierarchies for the overall properties of viscoelastic heterogeneous and composite materials, Arch. Mech. 47 (6), 1125–1155.
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.
Huet, C. (1999b), Coupled size and boundary-condition effects in viscoelastic heterogeneous and composite bodies, Mech. Mater. 31 (12), 787–829.
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.
Jeulin, D. (1997), Advances in Theory and Applications of Random Sets, World Scientific, Singapore.
Jeulin, D. (2001), this volume.
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.
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.
Kachanov, L.M. (1971), Foundations of the Theory of Plasticity, North-Holland, Amsterdam.
Kachanov, M. (1993), Elastic solids with many cracks and related problems, Adv. Appl. Mech. 30, 259–445.
Krajcinovic, D. (1996), Damage Mechanics, North-Holland, Amsterdam.
Kröner, E. (1972), Statistical Continuum Mechanics, CISM Lectures 92, Springer-Verlag, Wien.
Kröner, E. (1994), Nonlinear elastic properties of microheterogeneous media, ASME J. Eng. Mater. Tech. 116, 325–330.
Kruyt, N.P. and Rothenburg, L. (2001), Statics of the elastic behaviour of granular materials, Int. J. Solids Struct. 38, 4879–4889.
Lemaitre, J. and Chaboche, J.-L. (1994), Mechanics of Solid Materials,Cambridge University Press.
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.
Matheron, G. (1974), Random Sets and Integral Geometry,J. Wiley and Sons.
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. Jasiuk
D. Pamplona, C. Steele, I. Weber and L. Bevilacqua), Rio de Janeiro, Brazil, 369–372.
Noor, A.K. (1988), Continuum modeling for repetitive lattice structures, Appl. Mech. Rev. 41 (7), 285–296.
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.
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.
Ostoja-Starzewski, M. (1998), Random field models of heterogeneous materials, Intl. J. Solids Struct. 35 (19), 2429–2455.
Ostoja-Starzewski, M. (1999a), Scale effects in materials with random distributions of needles and cracks, Mech. Mater. 31 (12), 883–893.
Ostoja-Starzewski, M. (1999b), Microstructural disorder, mesoscale finite elements, and macroscopic response, Proc. R. Soc. Lond. A455, 3189–3199.
Ost.oja-Starzewski, M. (2000), Universal material property in conductivity of planar random microstructures, Phys. Rev. B62 (5), 2980–2983.
Ostoja-Starzewski, M. (2001), Microstructural randomness versus representative volume element in thermomechanics, ASME J. Appl. Mech.,in press.
Ostoja-Starzewski, M. (2001/02), Lattice models in rnicromechanics, Appl. Mech. Rev.,in press.
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.
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.
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.
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.
Ostoja-Starzewski, M. and Wang, C. (1989), Linear elasticity of planar Delaunay networks: Random field characterization of effective moduli, Acta Mech. 80, 61–80.
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.
Ponte Castaneda, P. and Suquet, P. (1998), Nonlinear composites, Adv. Appl. Mech. 34, 171–302.
Prohorov, Yu.V. and Rozanov, Yu.A. (1969), Probability Theory, Springer-Verlag, Berlin.
Pshenichnov, G. I. (1993), A Theory of Latticed Plates and Shells, World Scientific, Singapore.
Reuss, A. (1929), Berechnung der Fliessgrenze Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle, ZAMM 9, 49–58.
Rytov, S.M., Kravtsov, Yu.A., Tatarskii, V.I. (1987), Principles of Statistical Radiophysics, Springer-Verlag, Berlin.
Sah. K. (1991), Principe de Hill et homogénéisation des matériaux aléatoires, C. R. Acad. Sci. Paris II 312, 1–5.
Sab, K. (1992), On the homogenization and the simulation of random materials, Europ. J. Mech., A/Solids 11, 585–607.
Sab, K. (1994), Homogenization of nonlinear random media by a duality method. Application to plasticity, Asymptotic Anal. 9, 311–336.
Sachs, G. (1928), Zur Ableitung einer Fliessbedingung, Z. Ver. Deutsch. Ing. 72, 734–736.
Saigal, S. (1995), Stochastic boundary elements for two-dimensional potential flow in non-homogeneous media, Comp. Meth. Appl. Mech. Eng. 121, 211–230.
Sobczyk, K. (1985), Stochastic Wave Propagation, Elsevier-Polish Sci. Publ., Amsterdam-Warsaw.
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.
Stoyan, D., Kendall, W.S. and Mecke, J. (1987), Stochastic Geometry and its Applications,J. Wiley and Sons, New York.
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.
Suquet, P. ed. (1997), Continuum Micromechanics, Springer-Verlag, Wien.
Taylor, G.I. (1938), Plastic strain in metals, J. Inst. Met. 62, 307–324.
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.
Torquato, S. (1991), Random heterogeneous media: microstructure and improved bounds on effective properties, Appl. Mech. Rev. 44, 37–76.
Vanmarcke, E.H. (1983), Random Fields: Analysis and Synthesis, MIT Press, Cambridge, MA.
Vakulenko, A.A. and Kachanov, M.L. (1971), Continuum model of a medium with cracks (in Russian), Mech. Solids 4, 54–59.
Van Mier, J.G.M. (1997), Fracture Processes of Concrete, CRC Press, Boca Raton, FL.
Voigt; W. (1889), Über die Beziehung zwischen den beiden Elastizitätskonstanten isotroper Körper, Wied. Ann. 38 573–587.
Willis, J.R. (1981), Variational and related methods for the overall properties of composites, Adv. Appl. Mech. 21, 1–78.
Willis, J.R. and Talbot, D.R.S. (1989), in Prue . 6th Symp. Continuum Models and Discrete Systems (ed. G.A. Maugin ), Longmans, London.
Willis, J.R. (2001), this volume.
Wozniak, C. (1970), Surface Lattice Structures (in Polish), Polish Sci. Publ., Warsaw.
Yaglom, A.M. (1957), An Introduction to the Theory of Stationary Random Functions, Dover, New York.
Zubelewicz, A. and Bazant, Z.P. (1987), Interface modeling of fracture in aggregate composites, ASCE J. Eng. Mech. 113, 1619–1630.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Wien
About this chapter
Cite this chapter
Ostoja-Starzewski, M. (2001). Mechanics of Random Materials. In: Jeulin, D., Ostoja-Starzewski, M. (eds) Mechanics of Random and Multiscale Microstructures. International Centre for Mechanical Sciences, vol 430. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2780-3_3
Download citation
DOI: https://doi.org/10.1007/978-3-7091-2780-3_3
Publisher Name: Springer, Vienna
Print ISBN: 978-3-211-83684-2
Online ISBN: 978-3-7091-2780-3
eBook Packages: Springer Book Archive