Birkhäuser Boston, 2000
ISBN 0-8716-4083-5 ISBN 3-7643-4083-5 SPIN 10680933

Contents

1 Elementary Micromechanics of Heterogeneous Media
Konstantin Z. Markov

   1.1  Introduction
   1.2  The homogenization problem
   1.3  Some basic results
   1.4  The single inclusion problem
   1.5  One-particle approximations
   1.6  Elastic properties of polycrystals
   1.7  References

2  Diffusion-Absorption and Flow Processes in Disordered Porous Media
Salvatore Torquato

    2.1  Introduction
    2.2  Microstructure functions
    2.3  Steady-state trapping problem
    2.4  Time-dependent trapping problem
    2.5  Steady-state fluid permeability problem
    2.6  Time-dependent flow problem
    2.7  Variational principles for trapping problem
    2.8  Variational principles for flow problem
    2.9  Bounds on trapping constant
    2.10  Bounds on fluid permeability
    2.11  Cross-property relations
    2.12  References

3  Self-Consistent Methods in the Problem of Wave Propagation through Heterogeneous Media
Sergei K. Kanaun
    3.1  Introduction
    3.2  The main hypotheses of the methods
    3.3  Integral equation of the diffraction problem
    3.4  General scheme of the effective field method
    3.5  General scheme of versions I and II of the EMM for matrix composite materials
    3.6  The solutions of the one-particle problems of version I of the EMM and of the EFM
    3.7  Asymptotics of the solutions of the dispersion equations
    3.8  Versions II and III of the EMM in the case of spherical inclusions
    3.9  Version I of the EMM and the EFM in the case of isotropic random sets of inclusions
    3.10  Versions I, II, and III of the EMM for matrix composite materials
    3.11  An approximate solution of the one-particle problem
    3.12  The EFM for composites with regular lattices of spherical inclusions
    3.13  Versions I and IV for polycrystals and granular materials
    3.14  Discussion
    3.15  Conclusions
    3.16  References

4  Deformable Porous Media and Composites Manufacturing
  Angiolo Farina and Luigi Preziosi

    4.1  Introduction
    4.2  Ensemble average approach
    4.3  Effective media approach
    4.4  Deformable and saturated porous media models
    4.5  Boundary conditions
    4.6  One-dimensional infiltration
    4.7  Simulations
    4.8  Three-dimensional unsaturated isothermal model
    4.9  Open problems
    4.10  References

5  Micromechanics of Poroelastic Rocks
  Robert W. Zimmerman

   5.1  Introduction
   5.2  Hydrostatic poroelasticity
   5.3  Undrained compression
   5.4  Constitutive equations of linearized poroelasticity
   5.5  Equations of stress equilibrium and fluid flow
   5.6  Dependence of poroelastic parameters on pore structure
   5.7  Conclusions and future directions

Index

Heterogeneous Media: Modelling and Simulation

It is well known that almost all materials used in contemporary life and industry, both manufactured or occurring in nature, are inhomogeneous and multicomponent, possessing a rich and complicated internal structure. Appropriate examples can be cited from all branches of science, such as heterogeneous (composite) solids, mixtures and multicomponent fluids, soils and rocks and biological tissues. The internal structure, or the microstructure, plays a key role in understanding and controlling the macroscopical (continuum) behavior of such materials. In general, this is the micromechanics that takes as a basis a certain "microscopic picture" of the medium structure and then develops mathematical models and tools to predict the overall macroreaction, trying to take into account the appropriate microstructure. The so-obtained models and theories are tested in turn on realistic and typical examples and situations, explicit theoretical results are extracted either in analytical or numerical form, and a comparison with the experimental findings is performed. The degree of the observed coincidence between theory and experiments serves as an obvious test on the adequacy of both the microstructural "picture" and the subsequent modelling.

This general modelling scheme is certainly well known, having been repeated many times in many different contexts, including micromechanical studies of heterogeneous or multicomponent media. And this repetition brings us to one of the main goals of the present collection: In modelling and in the subsequent mathematical treatment, many micromechanical problems are either very close or share very similar basic ideas. These problems appear, however, in seemingly different contexts and amid different scientific disciplines (solid mechanics, hydromechanics, geophysics, solid state physics, diffusion-controlled reactions in chemical systems, biomechanics, etc.). Thus many diverse backgrounds, ways of thinking, and "languages" are used, and the relevant literature as a result is widely spread over journals possessing different styles and often mutually nonintersecting communities of readers. The ambitious aim of this book is just to alleviate this situation to a certain degree, through collecting several survey papers of actively working specialists and dealing with some of the most important problems in micromechanics of multicomponent systems, both from a theoretical and a practical viewpoint.

Contents

The contents are organized into five chapters.

The first chapter by Markov reviews the basic, introductory, and more elementary ideas and results of micromechanics of heterogeneous media. The central problem under discussion is "homogenization." It replaces such media by homogeneous ones, which behave macroscopically in the same way, and possess certain gross effective properties. These properties are related in a complicated manner to the prescribed internal structure of the medium, and their evaluation represents a profound challenge in any specific situation. A brief historical survey is given, underlying the reappearance of essentially the same "homogenization" quest in numerous guises and contexts over the last two centuries. Within the framework of the volume-averaging approach, the basic notions are introduced and some of the central, now classical, results are then derived and discussed-perturbation expansions, Hashin-Shtrikman's bounds, variational estimates and Levin's cross-property relation. A general "one-particle" scheme for approximate evaluation of the effective properties (in the static case) is detailed in its various implementations such as self-consistency, iterated limits and effective field. Illustrations concern conductivity, elasticity, and simplest absorption phenomena in heterogeneous media, as well as a simple self-consistent model for polycrystals' homogenization.

The rest of the chapters are more specialized, dealing in detail with various important phenomena in heterogeneous media and the peculiarities of their macroscopic modelling, based on appropriate microstructural descriptions.

The second chapter by Torquato is devoted to some rigorous methods for estimating effective properties associated with two different types of processes occurring in random porous media: diffusion-absorption and flow phenomena. The first problem, often referred to as the "trapping problem,'' examines the so-called trapping constant (or, equivalently, the mean survival time) and diffusion relaxation times. The second problem examines the fluid permeability, as well as the viscous relaxation times. The author reviews several topics: (i) microstructure characterization via statistical correlation functions; (ii) derivation of effective properties via homogenization theory; (iii) rigorous bounds on the effective properties in terms of correlation functions; and (iv) cross-property relations that rigorously link diffusion properties to flow properties.

The third chapter by Kanaun is concerned with the problems of evaluating mean wave fields and the effective dynamic properties of composite materials with random microstructure. The basic concepts of two of the main "self-consistent" schemes (the effective field and effective medium methods) and their application to these problems are reviewed and critically revisited. The main hypotheses of the methods do not depend on the types of propagating waves and hence can be employed to wave problems of different physical nature. The methods and their important modifications are developed for the case of monochromatic electromagnetic waves, propagating through particulate composite materials and polycrystals. Wide regions of variations in frequencies of the exciting fields and of physical properties of the composite constituents are considered. Predictions of the methods are compared with available experimental data and/or exact solutions in order to specify the areas of applicability of various implementations of the self-consistent schemes. The sources of possible inaccuracies of the methods and ways to overcome them are discussed.

The fourth chapter by Farina and Preziosi is devoted to modelling of the complicated processes used in real production technologies of modern composite materials. It is a very important subject since a growing number of industrial activities demands advanced materials that satisfy stringent requirements and lower costs. These requirements, which involve a combination of many properties, can often be satisfied by using a composite material, whose constituents act synergically to solve the needs of application. Modelling the behavior of such a heterogeneous material during its production is a very hard task, but it is very useful for the optimization of the manufacturing process itself. This chapter focuses on the deduction of appropriate mathematical models of deformable porous media and on their application to composite materials manufacturing as a first step toward the understanding of this complex process.

The fifth chapter by Zimmerman is a review of poroelasticity of rock-like media. It is developed here in a manner that assigns a central role to micromechanics, more specifically, to the micro-scale deformation of the pore space. A simplified theory that describes the deformation of porous media subjected to hydrostatic loading, under drained and undrained conditions, is first developed in a fully nonlinear form. The linearized theory of coupled mechanical deformation and fluid flow is then presented. The chapter concludes with a discussion of the effect of pore microstructure on the constitutive parameters of poroelasticity. Some further research directions are outlined as well.

Conclusion

The editors have made a sincere effort to collect in the present volume the different aspects, approaches, and results concerned with macroscopic modelling of heterogeneous media. The basic aim is to underline the common origin of the problems and the similarity of the basic ideas and their implementations, despite the widely different contexts and specific interpretations. Without any claim of completeness, we believe that the collection will contribute to mutual understanding between various scientific communities and will stimulate further and fruitful interdisciplinary research.

The editors also hope that any one of the surveys, included in the collection (maybe combined with one or two of the others), can be used as interdisciplinary introductory courses in various micromechanical topics for graduate and PhD students in applied and industrial mathematics.

Konstantin Z. Markov, Sofia, Bulgaria

Luigi Preziosi, Torino, Italy

           Contributors

Angiolo Farina
Università degli Studi di Firenze
Dipartimento di Matematica "U. Dini"
I-50134 Firenze, Italy
e-mail  visit@polito.it

Sergei K. Kanaun
Instituto Tecnologico y de Estudios Superiores de Monterrey
Campus Estado de México, Apd. Postal 6-3
Atizapan, Edo. de México
52926 México, México
e-mail  kanaoun@campus.cem.itesm.mx

Konstantin Z. Markov
Faculty of Mathematics and Informatics
"St. Kliment Ohridski" University of Sofia
5 blvd J. Bourchier, P.O.Box 48
BG-1164 Sofia, Bulgaria
e-mail  kmarkov@fmi.uni-sofia.bg

Luigi Preziosi
Dipartimento di Matematica
Politecnico di Torino
I-10129 Torino, Italy
e-mail  preziosi@polito.it

Salvatore Torquato
Princeton University
Princeton Materials Institute
and Mechanics, Materials & Structures Program
Princeton, New Jersey 08544, USA
e-mail  torquato@matter.princeton.edu

Robert W. Zimmerman
T.H. Huxley School of Environment,
Earth Sciences and Engineering
Imperial College of Science, Technology and Medicine
London SW7 2BP, UK
e-mail  r.w.zimmerman@ic.ac.uk