Table Of ContentModeling and Simulation in Science, Engineering and Technology
Series Editor
Nicola Bellomo
Politecnico di Torino
Italy
Advisory Editorial Board
M. Avellaneda S. Nikitin
Courant Institute of Mathematical Sciences Arizona State University
New York University USA
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H.G. Othmer
K.J. Bathe University of Utah
Massachusetts Institute of Technology USA
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V. Protopopescu
P. Degond CSMD
Universite P. Sabatier Toulouse 3 Oak Ridge National Laboratory
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Iowa State University Kyoto University
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P. Le Tallec E.S. $uhubi
INRIA, BP 105 Istanbul Technical University
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Heterogeneous Media
Micromechanics Modeling Methods
and Simulations
Konstantin Markov
L uigi Preziosi
Editors
Springer Science+Business Media, LLC
Konstantin Markov Luigi Preziosi
Faculty of Mathematics and Informatics Dipartimento di Matematica
University of Sofia Politecnico di Torino
St. Klimentohridski Torino 1-10129
Sofia BG-1164 Italy
Bulgaria
Library of Congress Cataloging-in-Publication Data
Heterogeneous media: micromechanics modeling methods and simulations /editors,
Konstantin Markov, Luigi Preziosi.
p. cm - (Modeling and simulat ion in science, engineering and technology)
Includes bibliographical references.
ISBN 978-1-4612-7098-0 ISBN 978-1-4612-1332-1 (eBook)
DOI 10.1007/978-1-4612-1332-1
1. Inhomogeneous materials-Mechanical properties.
2. Micromechanics-Mathematical models. 3. Porous materials-Mathematical methods.
4. Composite materials-Mathematical models. 1. Markov, Konstantin Z. II. Preziosi,
Luigi. III. Modeling and simulation in science, engineering and technology.
TA418.9.153 H48 1999
620.1'1299-dc21 99-046355
CIP
AMS Subject Classifications: 73B35, 73K20, 73E, 76S
Printed on acid-free paper.
© 2000 Springer Science+Business Media New York
Originally published by Birkhiiuser Boston in 2000
Softcover reprint of the hardcover 1s t edition 2000
AII rights reserved. This work may not be translated or copied in whole or in part without the
written permission of the publisher ( Springer Science+Business Media, LLC ), except for brief
excerpts in connection with reviews or scholarly analysis. Use in connection with any form of
information storage and retrieval, electronic adaptation, computer software, or by similar or
dissimilar methodology now known or hereafter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication, even
if the former are not especially identified, is not to be taken as a sign that such names, as
understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely
byanyone.
ISBN 978-1-4612-7098-0
Typeset by the editors in TeX.
9 8 7 6 5 4 3 2 1
Contents
Preface ix
Contributors xiii
1 Elementary Micromechanics of Heterogeneous Media 1
Konstantin Z. Markov
1.1 Introduction. . . . . . . . . . 2
1.2 The homogenization problem 21
1.3 Some basic results ............ 53
1.4 The single inclusion problem 85
1.5 One-particle approximations 105
1.6 Elastic properties of polycrystals 139
1.7 References . . . . . . . . . . . . . 146
2 Diffusion-Absorption and Flow Processes in Disordered
Porous Media 163
Salvatore Torquato
2.1 Introduction. 164
2.2 Microstructure functions . . . . 168
2.3 Steady-state trapping problem 172
2.4 Time-dependent trapping problem 175
2.5 Steady-state fluid permeability problem 181
2.6 Time-dependent flow problem ..... 185
2.7 Variational principles for trapping problem 187
2.8 Variational principles for flow problem 195
2.9 Bounds on trapping constant ....... . 202
v
VI Contents
2.10 Bounds on fluid permeability 215
2.11 Cross-property relations 224
2.12 References . . . . . . . . . . . 235
3 Self-Consistent Methods in the Problem of Wave
Propagation through Heterogeneous Media 241
Sergei K. Kanaun
3.1 Introduction.................. 242
3.2 The main hypotheses of the methods. . . . 245
3.3 Integral equation of the diffraction problem 250
3.4 General scheme of the effective field method . 253
3.5 General scheme of versions I and II of the EMM
for matrix composite materials . . . . . . 259
3.6 The solutions of the one-particle problems
of version I of the EMM and of the EFM 263
3.7 Asymptotics of the solutions of the
dispersion equations . . . . . . . . . . . . 267
3.8 Versions II and III of the EMM in the case of
spherical inclusions . . . . . . . . . . . . . . . 272
3.9 Version I of the EMM and the EFM in the case
of isotropic random sets of inclusions . 279
3.10 Versions I, II, and III of the EMM for
matrix composite materials .. 286
3.11 An approximate solution of the
one-particle problem . . . . . . . . . . . . . . 290
3.12 The EFM for composites with regular lattices
of spherical inclusions . . . . . . . . . 296
3.13 Versions I and IV for polycrystals and
granular materials 306
3.14 Discussion. 311
3.15 Conclusions 314
3.16 References. 315
4 Deformable Porous Media and
Composites Manufacturing 321
AngioID Farina and Luigi Preziosi
4.1 Introduction ........ . 322
4.2 Ensemble average approach 329
Contents
vii
4.3 Effective media approach ............. 341
4.4 Deformable and saturated porous media models . 352
4.5 BOlUldary conditions . . . . . 356
4.6 One-dimensional infiltration . . . . . . . . . . . . 363
4.7 Simulations . . . . . . . . . . . . . . . . . . . . . 374
4.8 Three-dimensional unsaturated isothermal model 385
4.9 Open problems 395
4.10 References. . . . . . . . . . . . . . . . 398
5 Micromechanics of Poroelastic Rocks 411
Robert W. Zimmerman
5.1 Introduction ....... . 411
5.2 Hydrostatic poroelasticity 413
5.3 Undrained compression . 421
5.4 Constitutive equations of linearized poroelasticity . 425
5.5 Equations of stress equilibrium and fluid flow 431
5.6 Dependence of poroelastic parameters on
pore structure . . . . . . . . . . . . 440
5.7 Conclusions and future directions . 459
5.8 References . 461
Index 471
Preface
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 inhomo
geneous and multicomponent, possessing a rich and complicated in
ternal 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 under
standing 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 over
all macroreaction, trying to take into account the appropriate mi
crostructure. 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 mi
cromechanical 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 treat
ment, many micromechanical problems are either very close or share
IX
x Preface
very similar basic ideas. These problems appear, however, in seem
ingly 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 nonintersect
ing 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 micro mechanics 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 heteroge
neous media. The central problem under discussion is "homogeniza
tion." 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 his
torical 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 conductiv
ity, elasticity, and simplest absorption phenomena in heterogeneous
media, as well as a simple self-consistent model for polycrystals' ho
mogenization.
The rest of the chapters are more specialized, dealing in detail
with various important phenomena in heterogeneous media and the
Preface
Xl
peculiarities of their macroscopic modelling, based on appropriate
microstructural descriptions.
The second chapter by Torquato is devoted to some rigorous meth
ods for estimating effective properties associated with two different
types of processes occurring in random porous media: diffusion
absorption and How 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 Huid 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 diffu
sion properties to How 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 con
cepts of two of the main "self-consistent" schemes (the effective field
and effective medium methods) and their application to these prob
lems 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 prop
erties 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 var
ious 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 require-
