Table Of ContentNORTH-HOLLAND SERIES IN
APPLIED MATHEMATICS
AND MECHANICS
EDITORS:
J.D. ACHENBACH
Northwestern University
B. BUDIANSKY
Harvard University
H.A. LAUWERIER
University of Amsterdam
P.G. SAFFIAN
California Institute of Technology
L. VAN WIJNGAARDEN
Twente Universtiy of Technology
J.R. WILLIS
University of Bath
VOLUME 37
1993
NORTH-HOLLAND-AMSTERDAM • LONDON • NEW YORK • TOKYO
MICROMECHANICS :
Overall Properties
of Heterogeneous Materials
by
S. NEMAT-NASSER
Department of Applied Mechanics and Engineering Sciences
University if California, San Diego
La Jolla, CA, U.S.A.
M. HORI
Department of Civil Engineering
University of Tokyo
Tokyo, Japan
1993
NORTH-HOLLAND-AMSTERDAM • LONDON • NEW YORK • TOKYO
ELSE VIER SCIENCE PUBLISHERS By.
P.O. Box 1991
1000 BZ Amsterdam
The Netherlands
Library of Congress Cataloging-in-Publication Data
Nemat-Nasser, S.
Mlcroeechanics : overall properties of heterogeneous saterlais /
by S. Neeat-Nasser, M. Hirt.
p. cm. -- (North-Holland series in applied iathe.attcs and
aechanics)
Includes bibliographical references and index.
ISBN 0-444-89881-6
1. Inhoiogeneous eaterials. 2. Micromechantcs. I. Hort, M.
(Muneo). II. Title. III. Series.
TA418.9.153146 1993
620.1'123--dc20 93-11093
CIP
ISBN: 0 444 89881 6
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PRINTED IN THE NETHERLANDS
PREFACE
This book consists of two parts. The first part deals with solids with
microdefects such as cavities, cracks, and inclusions, as well as with elastic
composites. The second part provides an introduction to the theory of linear
elasticity, added to make the book self-contained since linear elasticity serves as
the basis of the development of small-deformation micromechanics.
The material for the first part of this book grew out of lecture notes of the
first author in a course on micromechanics, which was initiated at the University
of California, San Diego (UCSD), in response to the existing need for a funda-
mental understanding of the micromechanics of the overall response and failure
modes of advanced materials, such as ceramics and ceramic and other compo-
sites. These advanced materials have become the focus of systematic and exten-
sive research at the Center of Excellence for Advanced Materials (CEAM),
which was established at UCSD in 1986, to include the U.S. Government's
University Research Initiative on Dynamic Performance of Materials. The
course, given in the Spring of 1987, was intended to furnish a basic background
for rigorous micromechanical modeling of the mechanical behavior and failure
regimes of a broad class of brittle materials.
Class notes taken by some of the graduate students were totally rewritten
by the second author in early 1988. Then this new version was completely
reworked by both authors, and used when the micromechanics course was given
for the second time in the Fall of 1988. A new version of the notes was then
completed and a major part of it was used again in the micromechanics course
in the Winter Quarter of 1990, and with more extensive revisions and additions,
again in the Winter Quarter of 1991. The manuscript was read by many gradu-
ate students who helped to correct misprints and related errors. The material
contained in this book as Part 1, is a thoroughly reexamined, modified,
expanded, and amended version of these class instruction notes. Included here
are new results on many basic issues in micromechanics, which we hope will be
helpful to graduate students and researchers dealing with rigorous physically-
based modeling of overall properties of heterogeneous solids; see Preface of
Part 1, for comments.
The second part, except for Sections 15 and 19, is essentially part of the
lecture notes on elasticity which the first author wrote in the late 1960's, while
teaching at UCSD. Many sections were done at that time, in collaboration with
the late Professor William Prager, and it was intended then to publish a mono-
graph on elasticity in a series of books in solid mechanics, by Blaisdell Publish-
ing Company. Unfortunately, before the monograph was completed, that series
was discontinued, and the authors were released from contractual obligations.
The notes were completed and used many times at UCSD and at Northwestern
University, for class instruction by the first author. The section on variational
PREFACE n1
methods included some new results of such generality that even today, much of
it is not known to many researchers. These were the results obtained in colla-
boration with Professor Prager. The variational principles in the current version,
given in Section 19, are even more general than those contained in the original
class notes. These principles include some new elements which should prove
useful for application to advanced modeling, as well as solutions of composites
and related heterogeneous bodies. Section 15 is a brief modern version of ele-
ments in vector and tensor algebra, and is particularly tailored to provide back-
ground for the rest of this book. Even for students of mechanics and mechanics
of materials familiar with the basic elements of linear elasticity, a quick exami-
nation of Subsection 15.5 should prove helpful for a better understanding of the
notation and some of the details of the manipulations involved in Part 1.
While the material in Part 2 is mostly standard, given for background
information, in preparing Part 1, the authors have benefited from the contribu-
tion of many modern researchers and numerous publications which have
appeared over the past few decades. Since the book is intended for use as a
graduate text, only a few selected references are cited within the text. Important
references may have been left out inadvertently, for which the authors apolo-
gize.
While the authors take full responsibility for any errors that remain, they
wish to express their appreciation to Mr. L. Ni and graduate students, B. Balen-
dran, Hang Deng, Mark Rashid, G. Subhash, John Wehrs, Niann-i Yu, Vinod
Sharma, Abbas Azhdari, Yeou-Fong Li, Anil Thakur, and Tomoo Okinaka, who
read various parts of various versions of the manuscript with care, and sought to
remove errors of minor or major importance. Thanks are also due John Willis
whose seminars and comments provided inspiration and insight, and led to con-
siderable improvement of several sections, as well as due Eva for proofreading
the entire manuscript, and Shiba for helping with word processing. The authors
also gratefully acknowledge the Army Research Office Contract DAAL-03-86-
K-0169 which provided the core support for the Center of Excellence for
Dynamic Performance of Materials at UCSD, and partially supported the two
authors during the preparation of this book. The book has been formatted by the
authors, using ditroff. Most of the figures, and all of the graphs have been con-
structed by pic and grap.
Sia Nemat-Nasser
Muneo Hon
La Jolla, California
September 17, 1992
PREFACE TO PART 1
Part 1 of this book contains a systematic development of the overall
response parameters of solids with microheterogeneities and defects such as
cracks and cavities. The work deals with small deformations, particularly
relevant to advanced materials such as ceramics and ceramic composites, as
well as metals and polymeric composites, in a deformation range where overall
geometrical dimensions and shapes are not altered substantially by material
deformation. While, for the most part, a linearly elastic matrix containing
linearly elastic inclusions or cavities and cracks, is considered, overall material
nonlinearity caused by microcrack formation and growth, for example, is
included. In addition, as discussed in Subsection 12.8 and in Appendix A of Part
1, the basic results apply directly to composites and heterogeneous solids con-
sisting of elastoplastic or rate-dependent elastoviscoplastic materials, as long as
the small-deformation theory applies, or small incremental steps are used.
Within the above-mentioned general framework, the subject matter of
micromechanics is treated in a deliberate and systematic manner, at each stage
beginning with the fundamentals which are then treated in depth with consider-
able care, leading to illustrative examples to bring out in a concrete fashion the
involved basic steps, and then providing a number of major results with broad
applicability.
More specifically, in Sections 1 and 2, Chapter I, the basic idea of a
heterogeneous representative volume element (RVE) is discussed. The associ-
ated boundary-value problem is formulated, both in terms of the rate of change
of the field variables, as well as in terms of the total quantities. Averaging
methods are examined in Section 2, and a series of important basic universal
results are presented. Essentially all (unless otherwise explicitly stated) the
results in Sections 1 and 2 are valid for small deformations of solids consisting
of any (elastic or inelastic) constituents with any material properties. These
comments apply to results presented in Subsections 2.1 to 2.5. In Subsection 2.5,
detailed discussions are given for solids consisting of material constituents
which admit (at the local level) stress and/or strain potentials. (The response,
however, need not be linear.) It is then shown how the overall macropotentials
relate directly to the average of the corresponding micropotentials. A set of
exact relations is obtained in this manner, and based on this, the notion of a
representative volume element is examined. The question of the effects of the
boundary data considered for an RIB, on the resulting overall energy density
(elastic but materially nonlinear), and on the overall effective moduli for a
linearly elastic RVE, is given a thorough examination, leading to a set of univer-
sal inequalities and two theorems which provide exact ordering relations when
uniform boundary tractions, linear boundary displacements, or general boundary
data are considered for an RIB. In particular, general results developed in Sub-
sections 2.5.6 and 2.5.7, relating to bounds on macropotentials and, therefore, on
4 PART1
the overall strain energy and complementary energy functionals, should prove
useful and be a guiding element for specific results outlined in later sections,
using the assumption (postulate) of statistical homogeneity, and employing sim-
ple models to calculate the local average quantities. Indeed, these universal
theorems are used to obtain rigorous, computable bounds of considerable gen-
erality in Section 9, as is discussed later on in the sequel.
In Subsection 2.6, questions of statistical homogeneity and representative
volume elements are again examined to clarify the nature of the boundary data
which may be assigned to an RVE, and their influence on the resulting average
stresses, strains, and their potentials. Conditions under which the average of the
product of the stress and strain (or their rates) equals the product of the
corresponding averages, are discussed.
Included in Sections 1 and 2 also is a brief discussion of nonmechanical
properties of inhomogeneous media. General results developed for mechanical
properties are specialized and applied to electrostatic, magnetostatic, thermal,
and diffusional properties of a heterogeneous RVE.
Chapter II is devoted to estimating, in a systematic manner, the overall
elasticity and compliance tensors of a linearly elastic matrix containing micro-
cavities and microcracks, using two simple models for the averaging procedure.
These are: the dilute distribution model which assumes that the inhomogeneities
are small and far apart, so that their interaction may be neglected; and the self-
consistent model which takes into account the corresponding interaction, in a
certain, overall, approximate manner In particular, in Section 3 the stress-strain
relations of linear elasticity are reviewed and the necessary background is pro-
vided for subsequent sections. In Section 4, a systematic discussion of the
overall stress and strain in a porous RIB is given for two limiting alternative
boundary data, namely, uniform tractions, and linear displacements. It is shown,
directly and in a simple manner, how the corresponding overall compliance and
elasticity tensors can be estimated, using the reciprocal theorem and simple esti-
mates of the cavity boundary displacements. These results are generally valid
for cavities of any shape or distribution, and, except for the assumption of linear
elasticity, no approximations are involved. In Section 5 the general results of
Section 4 are applied to porous, linearly elastic solids. The dilute-distribution
and the self-consistent models are used. The relation between these models is
discussed in terms of specific problems. Section 6 deals with elastic solids with
microcracks. Here again, the same two models are used in a number of illustra-
tive examples, and the corresponding results are compared and discussed. In
addition, a brief overview of recent advances in theoretical and experimental
evaluation of brittle failure in compression is presented in Subsection 6.9.
Chapter III is devoted to linearly elastic solids with elastic micro-
inclusions, as well as the elastic response of polycrystals. First, in Section 7, for
micro-inclusions of any geometry and elasticity, exact general expressions for
the overall elastic modulus and compliance tensors are obtained, for overall
uniform boundary tractions and overall linear boundary displacements, respec-
tively. Then in Subsection 7.3, the concepts of eigenstrain and eigenstress
required to homogenize the heterogeneous RVE are introduced and examined in
some detail. In particular, Eshelby's tensor for an ellipsoidal inclusion
PREFACE 5
embedded in a uniform, infinitely extended, linearly elastic solid, is presented,
together with its dual tensor (the first associated with an eigenstrain, and the
second associated with the corresponding eigenstress), their properties exam-
ined, together with their dual relations, and they are used to obtain consistency
conditions associated with the homogenization. These results are then related to
the H- and J-tensors, introduced in Section 4, to homogenize an elastic RVE
containing microcavities and microcracks. The results are then used to formulate
overall modulus and compliance tensors of an elastic RVE with elastic inclu-
sions, on the basis of the dilute-distribution and the self-consistent models; Sub-
sections 7.4 and 7.5. The formulation of the overall elasticity and compliance
tensors, in terms of the overall elastic energy of the RVE, is discussed in Sub-
section 7.6, focusing particularly on the required symmetry for the overall elasti-
city and compliance tensors. Section 8 contains specific illustrative examples for
elastic solids with micro-inclusions. A number of problems are worked out in
detail, and numerical illustrations are presented.
Upper and lower bounds for the overall elastic moduli are presented in
Section 9. First, the Hashin-Shtrikman variational principle, as generalized by
Willis, is presented, when either the eigenstrains or the eigenstresses are used to
homogenize the corresponding heterogeneous RVE, leading to two functionals:
one, when the overall uniform boundary tractions are prescribed, where the
eigenstrains are used for homogenization, and the other, when the overall linear
displacement boundary data are assigned, in which case the eigenstresses are
used to homogenize the RVE. This leads to an elegant dual principle, with the
corresponding Euler equations defining the associated consistency conditions.
The upper and lower bounds for the energy functionals are presented in
Subsection 9.2, and their generalization is given in Subsection 9.3. Direct esti-
mates of the overall moduli, using approximate correlation tensors, are
presented in Subsection 9.4. In Subsection 9.5, the Hashin-Shtrikman varia-
tional principle is generalized for boundary data other than uniform tractions
and linear displacements, and the corresponding generalized bounds are
obtained. With the aid of the universal theorems of Subsection 2.5.6, these
bounds are then related to the bounds for the uniform traction and linear dis-
placement boundary data. It is proved that two out of four possible approximate
expressions that result are indeed rigorous bounds. Explicit, computable, exact
upper and lower bounds for the overall moduli are then given, when the com-
posite is statistically homogeneous and isotropic. Finally, it is shown in Subsec-
tion 9.6 that these new observations lead to universal bounds on two overall
moduli of multi-phase composites, valid for any shape or distribution of phases.
Furthermore, it is established that the bounds are valid for any finite elastic solid
of ellipsoidal shape, consisting of any distribution of inhomogeneities of any
shape and elasticity. (In Section 13, it is proved that the same bounds emerge
for multi-phase composites with periodic, but otherwise completely arbitrary,
microstructure.)
For historical reasons, the bounds on the overall properties in Section 9
are based on the Hashin-Shtrikman variational principle. An alternative formu-
lation of exact computable bounds is to use the universal theorems of Subsection
2.5.6, together with proper choices of the reference elasticity or compliance
6 PART 1
tensors. This is presented in Subsection 9.5.6. It is also used in Subsection
9.7.2 to formulate bounds on parameters which define nonmechanical properties
(e.g., conductivity and resistivity tensors) of composites.
A number of averaging methods (models) are studied in a systematic
manner in Section 10. This includes the dilute-distribution and the self-
consistent methods, as well as the differential scheme, and the two- and three-
phase models. The double-inclusion method is discussed in Subsection 10.4,
together with the Mori-Tanaka result, which leads to a number of interesting
results; for example, the self-consistent estimate is shown to be a special case of
the double-inclusion model, and is related to the Hashin-Shtrikman bounds. The
double-inclusion model is then generalized to multi-inclusion models, where,
again, all the average field quantities are estimated analytically. For a set of
nested ellipsoidal regions of arbitrary aspect ratios and relative locations, which
is embedded in an infinitely extended homogeneous elastic solid of arbitrary
elasticity, and which undergoes transformations with uniform but distinct
transformation strains within each annulus, the resulting average strains of each
annulus are computed exactly and in closed form; the transformation strains in
the innermost region need not be uniform. Explicit results are presented for an
embedded double inclusion, as well as a nested set of n inclusions. As examples
of the application of the multi-inclusion model, a composite containing multi-
layer inclusions and a composite consisting of several distinct materials are con-
sidered, and their overall moduli are analytically estimated. Then, relations
among these approximate techniques are studied, and comments on other
averaging schemes are made.
The development of Eshelby's tensor in terms of the infinite-space Green
function, is contained in Section 11, where the properties of this tensor and its
dual are studied. Given in this section is the Mori-Tanaka result, and its general-
ization to the case when arbitrary nonuniform eigenstrains (or transformation
strains) are distributed in a region of arbitrary geometry which is contained in an
ellipsoidal domain which, in turn, is embedded in an infinite homogeneous
domain. This result provides a powerful tool for the study of, for example,
fiber-reinforced composites with coated and/or partially debonded fibers. Then,
relations among various average quantities are examined, and the energy associ-
ated with heterogeneity and, hence, the homogenizing eigenstrains or eigens-
tresses, is derived.
Chapter IV covers the fundamentals of heterogeneous elastic solids with
periodically distributed inhomogeneities, such as inclusions, fibers, cavities, and
cracks. This chapter includes a number of new results, while at the same time
presenting the theory in considerable depth, starting at an elementary level.
Section 12 provides background information and gives a number of illustrative
examples, including periodically distributed interacting cracks. For a periodic
structure, the concept of a unit cell is introduced and, using Fourier series, the
general solution is obtained. Specific classes of problems are then solved as
illustration, and the overall average elastic parameters are obtained in terms of
the geometry and properties of the representative unit cell. Application of the
general results to a unit cell with rate-dependent or rate-independent consti-
tuents is examined in Subsection 12.8.
PREFACE 7
Section 13 focuses on the overall response of solids with periodic micros-
tructure consisting of some arrangements of linearly elastic uniform micro-
inclusions embedded in a linearly elastic uniform matrix. First, an equivalent
homogeneous solid is defined by introducing suitable periodic eigenstrain or
eigenstress fields. Then, the Hashin-Shtrikman variational principle is applied
to solids with periodic microstructure, and bounds on the overall moduli are
obtained by defining energy functionals for the eigenstrain or eigenstress fields
in the equivalent homogeneous solid. The bounds for the periodic microstruc-
ture are exact and can be computed to any desired degree of accuracy. It is
shown in Subsection 13.5, that there are always two overall elastic parameters
whose bounds, obtained on the basis of the periodic and random (RVE) micros-
tructures, are identical, and hence exact. Moreover, these bounds are valid for
inclusions of any shape or elasticity. In addition, it is shown that, with the aid of
the minimum potential and complementary potential energies, bounds on the
overall parameters are obtained directly, by a suitable choice of a reference
elasticily or compliance tensor.
In Section 14 the concept of mirror images of points and vectors is intro-
duced and then used to decompose tensor-valued functions defined on the unit
cell, to their symmetric and antisymmetric parts. The decomposition is applied
to Fourier series representations of tensor-valued field quantities such as strain,
stress, and elastic moduli, resulting in considerable economy in numerical com-
putation and clarity in restrictions which must be imposed on the boundary data.
In Appendix A of Part 1, application of the basic results to nonlinear rate-
dependent and rate-independent inelastic heterogeneous solids is briefly exam-
ined. Illustrative constitutive relations for phenomenological and slip-induced
plasticity models are briefly presented, and their implementation in terms of the
general theories of the preceding sections is pointed out. First, certain rate-
independent phenomenological plasticity theories are outlined, with a brief
examination of slip-induced crystal plasticity. Then their interpretation in terms
of rate-dependent processes is mentioned.
