Micromechanics of Materials, with Applications

723 Pages·2018·19.547 MB·English

by  Mark Kachanov, Igor Sevostianov

Solid Mechanics and Its Applications
Mark Kachanov
Igor Sevostianov
Micromechanics
of Materials, with
Applications
Solid Mechanics and Its Applications
Volume 249
Series editors
J. R. Barber, Ann Arbor, USA
Anders Klarbring, Linköping, Sweden
Founding editor
G. M. L. Gladwell, Waterloo, ON, Canada
Aims and Scope of the Series
Thefundamentalquestionsarisinginmechanicsare:Why?,How?,andHowmuch?
The aim of this series is to provide lucid accounts written by authoritative
researchersgivingvisionandinsightinansweringthesequestionsonthesubjectof
mechanics as it relates to solids.
The scope of the series covers the entire spectrum of solid mechanics. Thus it
includes the foundation of mechanics; variational formulations; computational
mechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations
of solids and structures; dynamical systems and chaos; the theories of elasticity,
plasticity and viscoelasticity; composite materials; rods, beams, shells and
membranes; structural control and stability; soils, rocks and geomechanics;
fracture; tribology; experimental mechanics; biomechanics and machine design.
Themedianlevelofpresentationistothefirstyeargraduatestudent.Sometexts
aremonographs defining thecurrentstateofthe field; othersareaccessibletofinal
year undergraduates; but essentially the emphasis is on readability and clarity.
More information about this series at http://www.springer.com/series/6557
Mark Kachanov Igor Sevostianov
(cid:129)
Micromechanics of Materials,
with Applications
123
Mark Kachanov Igor Sevostianov
Department ofMechanical Engineering Department ofMechanical andAerospace
TuftsUniversity Engineering
Medford, MA NewMexico State University
USA LasCruces, NM
USA
ISSN 0925-0042 ISSN 2214-7764 (electronic)
Solid MechanicsandIts Applications
ISBN978-3-319-76203-6 ISBN978-3-319-76204-3 (eBook)
https://doi.org/10.1007/978-3-319-76204-3
LibraryofCongressControlNumber:2018933462
©SpringerInternationalPublishingAG,partofSpringerNature2018
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar
methodologynowknownorhereafterdeveloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom
therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor
for any errors or omissions that may have been made. The publisher remains neutral with regard to
jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations.
Printedonacid-freepaper
ThisSpringerimprintispublishedbytheregisteredcompanySpringerInternationalPublishingAG
partofSpringerNature
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Preface
Amodelshouldbeassimpleaspossiblebutnotsimpler.
AlbertEinstein
Micromechanics studies materials that are heterogeneous at microscale. They can
be both man-made (composites, metals, concrete, and foams) and naturally
occurring(crackedorporous rocks, bone, andice). Thegoal ofmicromechanicsis
to relate the physical behavior of such materials—in particular, their overall
(effective) properties—to the microstructure (geometric arrangement of the con-
stituents and their properties).
Developmentsinmicromechanicswerestarted byleadingscientists ofthetime,
such as Maxwell (in the context of effective conductivity of heterogeneous mate-
rials)andEinstein(theviscosityofsuspensions).Inthecontextofsolidmechanics,
developments started in 1950s–1960s, in particular, in works of Hill and Eshelby.
Amicromechanicshasnowgrownintoalargefield;withanumberofmonographs
and reviews published. We mention in particular the following books and collec-
tions of reviews: Christensen (1979), Mura (1987), Suquet (1997) Nemat-Nasser
and Hori (1999), Markov and Prziosi (2000), Torquato (2002), Milton (2004),
Dormieux, Condo, and Ulm (2006), Qu and Cherkaoui (2006), Kanaun and Levin
(2008), Böhm (2010), Dvorak (2013), Kushch (2013), and Kachanov and
Sevostianov (2013).
The developments appear to have been pursued by two different communities
resulting in two large branches of micromechanics:
(cid:129) Solid mechanics. Fundamental results on inhomogeneity problems, bounding
the effective properties, the mechanics of interactions have been obtained.
Substantial limitations, however, hinder their applications to materials science;
among them, geometries of inhomogeneities are too idealized (mostly, ellip-
soids); bounds are too wide and apply mostly to the isotropic materials;
(cid:129) Materials science. Applications to specific materials involve “irregular”
microstructures (inhomogeneity shapes do not resemble ellipsoids); the very
problem of their quantitative characterization is challenging. At the same time,
v
vi Preface
sufficiently simple relations that reflect specifics of the microstructure are nee-
ded. This has given rise to various empirical relations, their range of applica-
bility not always being clear.
The present book addresses this “weak link” and aims at narrowing the gap
between the two communities to the extent possible. It covers recent results on
quantitative modeling of “irregular” microstructures.
Chapters 2–6 give a thorough presentation of the theoretical fundamentals
supplemented, to some extent, by numerical studies. We note, to this end, that
computational studies play useful role in micromechanics: they allow direct veri-
fication of various theoretical models. They cannot, however, fully replace theo-
retical models: without them, predictions for those microstructures that have not
been examined computationally are not fully clear.
Chapter 7—the largest in the book—is devoted to applications of microme-
chanics to specific materials having complex microstructures (sprayed coatings,
bone, geomaterials, composites, foams, and ceramics). It demonstrates the possi-
bilities offered by micromechanics and, in particular, by some of the recent
advances.
The book is aimed at broad audience that includes the materials science com-
munity. The presentation is kept as simple as possible. A reader is expected to be
familiar with basics of continuum mechanics; a summary of background results is
given in Chap. 1.
Medford, USA Mark Kachanov
Las Cruces, USA Igor Sevostianov
Contents
1 Background Results on Elasticity and Conductivity . . . . . . . . . . . . . 1
1.1 Basic Equations of Linear Elasticity. Elastic Symmetries. . . . . . . 1
1.2 Energy Principles of Elasticity. . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.1 Virtual Changes of State. . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.2 The Principle of Virtual Displacements . . . . . . . . . . . . . 11
1.2.3 The Principle of Virtual Forces. . . . . . . . . . . . . . . . . . . 12
1.2.4 The Principle of Stationarity of Potential
Energy of an Elastic Solid . . . . . . . . . . . . . . . . . . . . . . 13
1.2.5 The Principle of Stationarity of Complementary
Energy of an Elastic Solid . . . . . . . . . . . . . . . . . . . . . . 15
1.3 Approximate Symmetries of the Elastic Properties . . . . . . . . . . . 16
1.4 A Summary of Algebra of Fourth-Rank Tensors. . . . . . . . . . . . . 22
1.4.1 Isotropic Fourth-Rank Tensors . . . . . . . . . . . . . . . . . . . 24
1.4.2 Anisotropic Fourth-Rank Tensors . . . . . . . . . . . . . . . . . 25
1.4.3 Transversely Isotropic Tensors . . . . . . . . . . . . . . . . . . . 26
1.4.4 Averaging of Tensors nn and nnnn Over Orientations
in Simplest Cases of Orientation Distribution. . . . . . . . . 28
1.4.5 Orthotropic Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.5 Thermal and Electric Conductivity: Fourier and Ohm’s Laws . . . 32
1.6 Green’s Tensors in Elasticity and Conductivity and Their
Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.6.1 General Representation of Green’s Tensor
in Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.6.2 Isotropic Elastic Material . . . . . . . . . . . . . . . . . . . . . . . 37
1.6.3 Transversely Isotropic Elastic Material . . . . . . . . . . . . . 38
1.6.4 Green’s Tensor for a Monoclinic Material,
in the Plane of Elastic Symmetry and in the Direction
Normal to It. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
1.6.5 Cubic Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
vii
viii Contents
1.6.6 Two-Dimensional Anisotropic Elastic Material. . . . . . . . 52
1.6.7 Derivatives of Green’s Tensor. . . . . . . . . . . . . . . . . . . . 56
1.6.8 Green’s Function in the Conductivity Problem. . . . . . . . 58
1.7 Dipoles, Moments, and Multipole Expansions in Elasticity and
Conductivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
1.7.1 System of Forces Distributed in Small Volume . . . . . . . 61
1.7.2 Dipole. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
1.7.3 Center of Dilatation . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
1.7.4 Force Couple. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
1.7.5 Center of Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
1.7.6 Multipole Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 66
1.8 Stress Intensity Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
1.9 General Thermodynamics Framework for Transition from
Microscale to Macroscopic Constitutive Equations
(Rice’s Formalism) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
1.10 MathematicalAnalogiesBetweenElastostaticsandSteady-State
Heat Flux. Conductivity Analogues of Stress
Intensity Factors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
1.11 Discontinuities of the Elastic and Thermal Fields at Interfaces
of Two Different Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
1.11.1 Stress Discontinuities in the Elasticity Problem . . . . . . . 83
1.11.2 Flux Discontinuities in the Conductivity Problem. . . . . . 86
2 Quantitative Characterization of Microstructures in the Context
of Effective Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
2.1 Representative Volume Element (RVE) and Related Issues. . . . . 90
2.1.1 Hill’s Condition. Homogeneous Boundary
Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
2.1.2 Averages Over Volume and Their Relation to
Quantities Accessible on Its Boundary. . . . . . . . . . . . . . 93
2.1.3 Volumes Smaller than RVE . . . . . . . . . . . . . . . . . . . . . 98
2.2 The Concept of Proper Microstructural Parameters . . . . . . . . . . . 103
2.3 The Simplest Microstructural Parameters and Their
Limitations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
2.4 Microstructural Parameters Are Rooted in the Non-interaction
Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
2.5 Property Contribution Tensors of Inhomogeneities . . . . . . . . . . . 110
2.6 Hill’s Comparison (Modification) Theorem
and Its Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
2.7 Microstructural Parameters Are Different for Different
Physical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
2.8 Benefits of Identifying the Proper Microstructural
Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Contents ix
2.9 On the “Fabric” Tensor Approach . . . . . . . . . . . . . . . . . . . . . . . 121
2.10 Summary on Microstructural Characterization. . . . . . . . . . . . . . . 125
3 Inclusion and Inhomogeneity in an Infinite Space
(Eshelby Problems). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.1 The First and the Second Eshelby Problems. . . . . . . . . . . . . . . . 127
3.1.1 The Eigenstrain Problem
(The First Eshelby Problem). . . . . . . . . . . . . . . . . . . . . 128
3.1.2 The Inhomogeneity Problem
(The Second Eshelby Problem). . . . . . . . . . . . . . . . . . . 133
3.1.3 Eshelby Theorem for the Ellipsoidal Domain. . . . . . . . . 134
3.1.4 Extension of the Eshelby Theorem to Nonlinear
Ellipsoidal Inhomogeneities . . . . . . . . . . . . . . . . . . . . . 138
3.2 Elastic Fields Outside Inhomogeneities and Inclusions . . . . . . . . 141
3.2.1 Stress Concentrations at Boundary of an
Inhomogeneity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
3.2.2 External Fields in the Inclusion (Eigenstrain)
Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
3.2.3 Connection Between Inclusion- and Inhomogeneity-
Generated Elastic Fields . . . . . . . . . . . . . . . . . . . . . . . . 145
3.2.4 Far-Field Asymptotics of Elastic Fields of
Inhomogeneities and Its Relation to the Effective
Elastic Properties. The Multipole Expansion . . . . . . . . . 146
3.2.5 Shape Dependence of the Far-Field: Inhomogeneity
VersusInclusion.Far-FieldofanInclusionofArbitrary
Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
3.3 Ellipsoidal Inhomogeneities and Inclusions in the Isotropic
Matrix: Special Cases of Ellipsoid Geometry . . . . . . . . . . . . . . . 155
3.4 Spheroidal Inhomogeneity Embedded in a Transversely
Isotropic Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
3.5 Non-ellipsoidal Inclusion in Isotropic Material
(First Eshelby Problem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
3.5.1 Eshelby Tensor for a Cuboid (Rectangular
Parallelepiped) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
3.5.2 Eshelby Tensor for Polyhedra. . . . . . . . . . . . . . . . . . . . 173
3.5.3 Eshelby Tensor for a Supersphere. . . . . . . . . . . . . . . . . 174
3.5.4 Eshelby Tensor for a Torus. . . . . . . . . . . . . . . . . . . . . . 176
3.6 Eshelby Problem for Conductivity . . . . . . . . . . . . . . . . . . . . . . . 178
3.6.1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . 179
3.6.2 Ellipsoidal Inhomogeneity . . . . . . . . . . . . . . . . . . . . . . 181
3.6.3 Ellipsoidal Inhomogeneity in an Isotropic Matrix. . . . . . 182
3.6.4 Ellipsoidal Inhomogeneity Arbitrarily Oriented
in an Orthotropic Matrix. . . . . . . . . . . . . . . . . . . . . . . . 183