ft OF ELASTIC AND INELASTIC SOLIDS
Toshio Mura
Micromechanics
of Defects in Solids
Second, Revised Edition
MARTINUS NHHOFF PUPL1SHER5

MECHANICS OF ELASTIC AND INELASTIC SOLIDS
Editors: S. Nemat-Nasser and G.JE. Oravas
G.M.L. Gladwell, Contact problems in the classical theory of elasticity.
1980. ISBN 90-286-0440-4.
G. Wempner, Mechanics of solids with applications to thin bodies. 1982.
ISBN 90-286-0880-X.
T. Mura, Micromechanics of defects in solids. Second, revised edition,
1987. ISBN 90-247-3343-X (HB), and ISBN 90-247-3256-5 (PB).
R.G. Payton, Elastic wave propagation in transversely isotropic media.
1983. ISBN 90-247-2843-6.
S. Nemat-Nasser, H. Abe, S. Hirakawa, Hydrawlic fracturing and geo-
thermal energy. 1983. ISBN 90-247-2855-X.
S. Nemat-Nasser, R.J. Asaro, G.A. Hegemier (eds.), Theoretical founda-
foundation for large-scale computations for nonlinear material behavior. 1984.
ISBN 90-247-3092-9.

Micromechanics of
defects in solids
Second, revised edition
Toshio Mura
Department of Civil Engineering
Northwestern University, Evanston, IL, U.S.A.
1987 MARTINUS NIJHOFF PUBLISHERS
a member of the KLUWER ACADEMIC PUBLISHERS GROUP
DORDRECHT / BOSTON / LANCASTER

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P.O. Box 322, 3300 AH Dordrecht, The Netherlands
Library of Congress Cataloging in Publication Data
Mura, Toshio, 1925-
Micromcchanics of defects in solids.
(Mechanics of elastic and inelastic solids : 3)
Bibliography, p.
Includes indexes.
1. Micromechartics. 2. Solids. 3. Crystals—Defects.
I. Title. II. Series.
QC176.8.M5M87 1986 620.1'05 85-29652
ISBN 90-247-3256-5
ISBN 90-247-3343-X
Book Information
First edition 1982.
Copyright
© 1987 by Martinus Nijhoff Publishers, Dordrecht.
All rights reserved. No part of this publication may be reproduced, stored in a
retrieval system, or transmitted in any form or by any means, mechanical,
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the publishers,
Martinus Nijhoff Publishers, P.O. Box 163, 3300 AD Dordrecht,
The Netherlands.
PRINTED IN THE NETHERLANDS

Preface
This book stems from a course on Micromechanics that I started about fifteen
years ago at Northwestern University. At that time, micromechanics was a
rather unfamiliar subject. Although I repeated the course every year, I was
never convinced that my notes have quite developed into a final manuscript
because new topics emerged constantly requiring revisions, and additions. I
finally came to realize that if this is continued, then I will never complete the
book to my total satisfaction. Meanwhile, T. Mori and I had coauthored a
book in Japanese, entitled Micromechanics, published by Baifu-kan, Tokyo, in
1975. It received an extremely favorable response from students and re-
researchers in Japan. This encouraged me to go ahead and publish my course
notes in their latest version, as this book, which contains further development
of the subject and is more comprehensive than the one published in Japanese.
Micromechanics encompasses mechanics related to microstructures of
materials. The method employed is a continuum theory of elasticity yet its
applications cover a broad area relating to the mechanical behavior of materi-
materials: plasticity, fracture and fatigue, constitutive equations, composite materi-
materials, polycrystals, etc. These subjects are treated in this book by means of a
powerful and unified method which is called the 'eigenstrain method.' In
particular, problems relating to inclusions and dislocations are most effectively
analyzed by this method, and therefore, special emphasis is placed on these
topics. When this book is used as a text for a graduate course, Sections 3, 11,
and 22 should be emphasized.
Eigenstrain is a generic name given by the author to such nonelastic strains
as thermal expansion, phase transformation, and misfit strains. J.D. Eshelby,
who is a pioneer in this area, refers to eigenstrains as stress-free transforma-
transformation strains in his celebrated papers A957,1959). The term eigenstrain should
not be confused with the term 'eigenvalue' which occurs in mathematical
physics, and relates to an entirely different concept.
No particular background is required of readers of this book because
necessary mathematics and physics are explained in the text and Appendix.

vi Preface
Although I have tried to be fair in citing literature, I have to apologize if some
papers do not receive proper credit or are not cited.
The sections and subsections marked with an asterisk (*), can be skipped in
the first reading, since the subjects discussed there are peripheral to the main
theme.
I wish to express my thanks to all the people who have helped me during
the course of the preparation of the manuscript: my previous graduate
students, Zissis A. Moschovidis, Minoru Taya, Carl R. Vilmann, and Ronald
B. Castles, as well as my friends R. Furuhashi, N. Kinoshita, T. Morita, M.
Inokuti, and T. Mori. Mori receives my special thanks for having advised me
on the subject matter, for discussing with me whole chapters, and for helping
me to write Chapter 7. The manuscript reached the final form in his hands. I
also wish to thank S. Nemat-Naser who has read through the manuscript and
has given valuable comments.
I give my thanks to Vera Fisher for her skillful typing and her great
patience with me, to the secretaries whom I involved in various aspects of the
work over the years: Erika Ivansons, Miriam Littell, Masa Sumikura, and
Carolyn Andrews, and to my family for their patience and understanding.
Finally, I acknowledge the National Science Foundation and the U.S. Army
Research Office for their support of my research in the area of micromecha-
nics.
November 13,1980 T.M.

Contents
Preface v
Chapter 1. General theory of eigenstrains 1
1. Definition of eigenstrains 1
2. Fundamental equations of elasticity 3
Hooke's law 3
Equilibrium conditions 5
Compatibility conditions 6
3. General expressions of elastic fields for given eigenstrain
distributions 7
Periodic solutions 7
Method of Fourier series and Fourier integrals 9
Method of Green's functions 11
Isotropic materials 13
Cubic crystals 14
Hexagonal crystals (transversely isotropic) 14
4. Exercises of general formulae 15
A straight screw dislocation 15
A straight edge dislocation 18
Periodic distribution of cuboidal precipitates 20
5. Static Green's functions 21
Isotropic materials 22
* Anisotropic materials 25
* Transversely isotropic materials 26
* Krdner's formula 31
* Derivatives of Green's functions 32
* Two-dimensional Green's function 34
6. Inclusions and inhomogeneities 38
Inclusions 38
Inhomogeneities 40
* Effect of isotropic elastic moduli on stress 42
vii

viii Contents
7. Dislocations
Volterra and Mura formulas
* The Indenbom and Orlov formula
* Disclinations
8. Dynamic solutions
Uniformly moving edge dislocation
Uniformly moving screw dislocation
* 9. Dynamic Green's functions
Isotropic materials
Steady State
*10. Incompatibility
* Riemann-Christoffel curvature tensor
Chapter 2. Isotropic inclusions
11. Eshelby's solution
Interior points
Sphere
Elliptic cylinder
Penny-shape
Flat ellipsoid
Oblate spheroid
Prolate spheroid
Exterior points
Thermal expansion with central symmetry
*12. Ellipsoidal inclusions with polynomial eigenstrains
* The I-integrals
* Sphere
* Elliptic cylinder
* Oblate spheroid
* Prolate spheroid
* Elliptical plate
* The Ferrers and Dyson formula
13. Energies of inclusions
Elastic strain energy
Interaction energy
Strain energy due to a spherical inclusion
Elliptic cylinder
Penny-shaped flat ellipsoid
Spheroid
*14. Cuboidal inclusions
44
45
48
49
53
55
57
57
61
64
65
71
74
74
75
79
80
81
83
84
84
84
88
89
92
93
94
94
94
95
95
97
97
99
101
101
101
102
104

Contents ix
15. Inclusions in a half space 110
Green's functions 110
Ellipsoidal inclusion with a uniform dilatational eigen-
strain 114
* Cuboidal inclusion with uniform eigenstrains 121
* Periodic distribution of eigenstrains 121
Joined half-spaces 123
Chapter 3. Anisotropic inclusions 129
16. Elastic field of an ellipsoidal inclusion 129
17. Formulae for interior points 133
Uniform eigenstrains 134
Spheroid 137
Cylinder (elliptic inclusion) 141
Flat ellipsoid 143
Eigenstrains with polynomial variation 144
Eigenstrains with a periodic form 144
*18. Formulae for exterior points 149
Examples 156
19. Ellipsoidal inclusions with polynomial eigenstrains in aniso-
anisotropic media 158
Special cases 160
*20. Harmonic eigenstrains 161
21. Periodic distribution of spherical inclusions 165
Chapter 4. Ellipsoidal inhomogeneities 177
22. Equivalent inclusion method 178
Isotropic materials 181
Sphere 183
Penny shape 184
Rod 185
Anisotropic inhomogeneities in iso tropic matrices 187
Stress field for exterior points 187
23. Numerical calculations 188
Two ellipsoidal inhomogeneities 192
*24. Impotent eigenstrains 198
25. Energies of inhomogeneities 204
Elastic strain energy 204
Interaction energy 208
Colunneti's theorem 211
Uniform plastic deformation in a matrix 213
Energy balance 215

x Contents
26. Precipitates and martensites 218
Isotropic precipitates 219
Anistropic precipitates 220
Incoherent precipitates 226
Martensitic transformation 229
Stress orienting precipitation 237
Chapter 5. Cracks 240
27. Critical stresses of crakes in isotropic media 240
Penny-shaped cracks 240
Slit-like cracks 242
Flat ellipsoidal cracks 244
Crack opening displacement 247
28. Critical stresses of cracks in anisotropic media 248
Uniform applied stress 248
Non-uniform applied stress 253
* II integrals for a penny-shaped crack 255
* II integrals for cubic crystals 255
* II integrals for transversely isotropic materials 257
29. Stress intensity factor for a flat ellipsoidal crack 260
Uniform applied stresses 264
Non-uniform applied stresses 268
30. Stress intensity factor for a slit-like crack 271
Uniform applied stresses 272
Non-uniform applied stresses 274
Isotropic materials 274
31. Stress concentration factors 277
Simple tension 278
Pure shear 279
32. Dugdale-Barenblatt cracks 280
BCS model 288
Penny shaped crack 292
*33. Stress intensity factor for an arbitrarily shaped plane crack 297
* Numerical examples 305
34. Crack growth 307
Energy release rate 307
The J-integral 311
Fatigue 314
Dynamic crack growth 319

Contents xi
Chapter 6. Dislocations 324
35. Displacement fields 324
Parallel dislocations 325
A straight dislocation 327
36. Stress fields 327
Dislocation segments 328
Willis' formula 333
The Asaro et al. formula 334
Dislocation loops 335
37. Dislocation density tensor 338
Surface dislocation density 341
Impotent distribution of dislocations 343
38. Dislocation flux tensor 345
Line integral expression of displacement and plastic dis-
distortion fields 348
The elastic field of moving dislocationswave equations of
tensor potentials. 351
Wave equations of tensor potentials 352
39. Energies and forces 353
Dynamic consideration 354
40. Plasticity 361
Mathematical theory of plasticity 361
Dislocation theory 363
Plane strain problems 365
Beams and cylinders 373
41. Dislocation model for fatigue crack initiation 379
Chapter 7. Material properties and related topics 388
42. Macroscopic average 388
Average of internal stresses 388
Macroscopic strains 389
Tanaka-Mori's theorem 390
Image stress 393
Random distribution of inclusions-Mori and Tanaka's
theory 394
43. Work-hardening of dispersion hardened alloys 398
Work-hardening in simple shear 398
Dislocations around an inclusion 402
Uniformity of plastic deformation 405
44. Diffusional relaxation of internal and external stresses 406
Relaxation of the internal stress in a plastically deformed
dispersion strenthened alloy 407

xii Contents
Diffusional relaxation process, climb rate of an Orowan
loop 408
Recovery creep of a dispersion strengthened alloy 412
Interfacial diffusional relaxation 414
45. Average elastic moduli of composite materials 421
The Voigt approximation 421
The Reuss approximation 424
Hill's theory 426
Eshelby's method 428
Self-consistent method 430
Upper and lower bounds 433
Other related works 437
46. Plastic behavior of polycrystalline metals and composites 439
Taylor's analysis 439
Self-consistent method 443
Embedded weakened zone 448
47. Viscoelasticity of composite materials 449
Homogeneous inclusions 449
Inhomogeneous inclusions 452
Waves in an infinite medium 453
48. Elastic wave scattering 455
Dynamic equivalent inclusion method 459
Green's formula 460
49. Interaction between dislocations and inclusions 463
Inclusions and dislocations 463
Cracks in two-phase materials 471
50. Eigenstrains in lattice theory 477
A uniformly moving screw dislocation 480
51. Sliding inclusions 484
Shearing Eigenstrains 486
Spheroidol inhomogeneous inclusions 488
52. Recent developments 492
Inclusions, precipitates, and composites 492
Half-spaces 494
Non-elastic matnces 494
Cracks and inclusions 495
Sliding and debonding inclusions 497
Dynamic cases 497
Miscellaneous 498

Contents xiii
Appendix 1
Einstein summation convention
Kronecker delta
Permutation tensor
Appendix 2
The elastic moduli for isotropic materials
Appendix 3
Fourier series and integrals
Dirac's delta function and Heaviside's unit function
Laplace transform
Appendix 4
Dislocations pile-up
References
Author index
Subject index
499
499
499
499
501
501
505
505
507
508
510
510
513
572
582

General theory of eigenstrains
The definition of eigenstrains is given first. Then the associated general
solutions for elastic fields for given eigenstrains are expressed by Founer
integrals and Green's functions. Some details of calculations for Green's
functions are described for static and dynamic cases.
As fundamental formulae for the subsequent chapters, general expressions
of elastic fields are given for inclusions, dislocations, and disclinations. The
stress discontinuity on boundaries of inclusions and the incompatibility of
eigenstrains are discussed as general theories.
Throughout this work, a fixed rectangular Cartesian coordinate system with
coordinate axes xo i = 1, 2, 3, is used.
1. Definition of eigenstrains
'Eigenstrain' is a generic name given by the author to such nonelastic strains
as thermal expansion, phase transformation, initial strains, plastic strains, and
misfit strains. 'Eigenstress' is a generic name given to self-equilibrated internal
stresses caused by one or several of these eigenstrains in bodies which are free
from any other external force and surface constraint. The eigenstress fields are
created by the incompatibility of the eigenstrains.
This new English terminology was adapted from the German 'Eigenspan-
nungen und Eigenspannungsquellen,' which is the title of H. Reissner's paper
A931) on residual stresses. Eshelby A957) referred to eigenstrains as stress-free
transformation strains in his celebrated paper which has stimulated the present
author to work on inclusion and dislocation problems. The term 'elastic
polarization' was used by Kroner A958) for eigenstrains in a slightly different
context—when the nonhomogeneity of polycrystal deformation is under con-
consideration.
Engineers have used the term 'residual stresses' for the self-equilibrated
internal stresses when they remain in materials after fabrication or plastic
deformation. Eigenstresses are called thermal stresses when thermal expansion

Chap 1 General theory of eigenstrains
Fig 1 1 Inclusion п
is a cause of the corresponding elastic fields. For example, when a part ?2 of a
material (Fig. 1.1) has its temperature raised by T, thermal stress a is
induced in the material D by the constraint from the part which surrounds ?2.
The thermal expansion aT, where a is the linear thermal expansion coefficient,
constitutes the thermal expansion strain,
**j = suaT> AЛ)
where 5,y is the Kronecker delta (see Appendix 1). The thermal expansion
strain is the strain caused when ?2 can be expanded freely with the removal of
the constraint from the surrounding part.
The actual stram is then the sum of the thermal and elastic strains. The
elastic strain is related to the thermal stress by Hooke's law. The thermal
expansion strain A.1) is a typical example of an eigenstrain. In the elastic
theory of eigenstrains and eigenstresses, however, it is not necessary to
attribute e* to any specific source. The source could be phase transformation,
precipitation, plastic deformation or a fictitious source necessary for the
equivalent inclusion method (to be discussed in Section 22).
When an eigenstrain e* is prescribed in a finite subdomain ?2 in a
homogeneous material D (see Fig. 1.1) and it is zero in the matrix D-?2, then
?2 is called an inclusion. The elastic moduli of the material are assumed to be
homogeneous when inclusions are under consideration.
If a subdomain ?2 in a material D has elastic moduli different from those of
the matrix, then ?2 is called an inhomogeneity. Applied stresses will be
disturbed by the existence of the inhomogeneity. This disturbed stress field
will be simulated by an eigenstress field by considering a fictitious eigenstrain
tfj in ?2 in a homogeneous material.
When ?2 in Fig. 1.1 is a plane embedded in a three-dimensional material D
and e*j is given on ?2 as a plastic strain caused by a finite slip b, the boundary

2 Fundamental equations of elasticity 3
of Й is called a dislocation loop. If e*y. is created by a rigid rotation of plane ft
by ы, the boundary of Q is called a disclination loop.
2. Fundamental equations of elasticity
In this section the field equations for the elasticity theory will be reviewed with
particular reference to solving eigenstrain problems. These problems consist of
finding displacement ut, strain ttj, and stress atj at an arbitrary point
x(xv x2, x3) when a free body D is subjected to a given distribution of
eigenstrain e*j. A free body is one which is free from any external surface or
body force.
Hooke's law
For infinitesimal deformations considered in this book, the total strain с. is
regarded as the sum of elastic strain ei} and eigenstrain e* ,
The total strain must be compatible,
where и. ^ = Зм,/3ху.
The elastic strain is related to stress atJ by Hooke's law;
au = c,jkieu = cijki(*ki ~ «*/) B-3)
or
°.v=Cy*/K/-?2/)> B-4)
where С,.у-А/ are the elastic moduli (constants) (see Appendix 2), and the
summation convention for the repeated indices is employed (see Appendix 1).
Since Cljkl is symmetric (C,jlk = CtJkl), we have CijklulJc = Cijklukl. In the
domain where e* = 0, B.4) becomes
% = cijkfiki=cijkiuk,i- B-5)
The inverse expression of B.3) is
%~^=С-1к1ок„ B.6)
where C~l, is the elastic compliance.

4 Chap. 1 General theory of eigenstrains
For isotropic materials, B.3) and B.6) can be written as
a,-, = M(l7?,),,(tJt,),
where Л and ju are the Lame constants, and v is Poisson's ratio. Young's
modulus E, the shear modulus ц, and the bulk modulus К are connected by
2ц = ?"/A + p), K = E/3A - 2v), and Л = 2/л»/A - 2р). The alternative ex-
expressions for B.7) are
лу 1 + V
=—(
Е (
and
B-8)
~]Г°Ху> B-9)
zx ?j «'

2 Fundamental equations of elasticity 5
where ekk = ex + ey + ez and <.*kk = ?* + ?* + e*. It is convenient to use B.8) for
the plane strain case where ez = 0. Expression B.9) is recommended for the
plane stress case where a, = azx = azy = 0. It should be noted that solutions for
the plane stress can be obtained directly from those for the plane strain by
replacing E/{\ - v1) with E and v/(l - v) with v.
When Hooke's law B.8) is rewritten for the two-dimensional case, we have
B.9.1)
°z = °zx = °zy -
for the plane stress and к = C - v)/{l + v). For the plane strain, we have
3 - к
where к = 3 - 4v.
Equilibrium conditions
When eigenstresses are calculated, material domain D is assumed to be free
from any external force and any surface constraint. If these conditions for the
free body are not satisfied, the stress field can be constructed from the
superposition of the eigenstress of the free body and the solution of a proper
boundary value problem.
The equations of equilibrium are
%,7 = 0 (i = l,2,3). B.10)

6 Chap. 1 General theory of eigenstrains
The boundary conditions for free external surface forces are
°ij"j = 0, B-П)
where nt is the exterior unit normal vector on the boundary of D.
By substituting B.4) into B.10) and B.11), we have
Cijk,»k,,j = CijkAuj B-12)
and
cijkiukjnj = cijki4inj- B-13)
It can be seen that the contribution of t* to the equations of equilibrium is
similar to that of a body force since the equations of equilibrium under body
force Xt with zero e* are Cijkluklj = — Xt. Similarly, Cijklt*klrij behaves like a
surface force on the boundary. Thus, it can be said that the elastic displace-
displacement field caused by e*, in a free body is equivalent to that caused by body
force -Cijkle*klj and surface force Cijklt*klnf
In subsequent chapters, D in most cases is considered as an infinitely
extended body (infinite body), and condition B.11) is replaced by the condi-
condition otJ{x) -> 0 for x -* oo.
Compatibility conditions
The strain tensor ttJ has six components, while the displacement vector ut has
three components. The tensor and the vector are related to each other through
the relation B.2), which can be called the condition for the compatibility of
strain f.tj. Generally, however, the equations of compatibility are referred to
the relations which are derived from B.2) by eliminating ut,
where epki is the permutation tensor (see Appendix 1). Relation B.14) will be
discussed in Section 10.
The displacement differential equations of the elasticity theory are given by
B.12). In some cases, however, it is more convenient to consider B.10), B.3),
and B.14). Boundary conditions and various side conditions, such as singular-
singularity conditions, continuity conditions, etc., arise in problems from time to time.
We can say at this point that the fundamental equations to be solved are
equations B.12).

S. General expressions of elastic fields for given eigenstrain distributions 7
Eigenstresses are caused by constraint from the surrounding elastic medium
which prohibits the geometrically incompatible deformation of c*. The incom-
incompatibility of c* was discussed by Reissner A931) and Nemenyi A931).
Dislocations due to incompatibility were studied by Weingarten A901), Cesaro
A906), Volterra A907), and Moriguti A947) from the viewpoint of the
elasticity theory in connection with the multiple values of displacements and
rotations. Another viewpoint on dislocations, from the plasticity theory, was
developed by Kondo A955), Bilby A960), and Kroner A958).
In the following sections we investigate the methods of finding the associ-
associated elastic fields (displacements, strains, stresses) and the related problems
for given distributions of e*. Particular emphasis will be placed on the case
when a uniform e* is given in an ellipsoidal domain Q in an infinitely
extended medium D. The results are useful for the study of the mechanical
properties of solids which may contain precipitates, inclusions, voids, cracks,
etc. The most fundamental contribution to this study was made by Eshelby
A951, 1956, 1957, 1959 and 1961).
3. General expressions of elastic fields for given eigenstrain distributions
The case where a given material is infinitely extended is of particular interest
for the mathematical simplicity of the solution as well as for its practical
importance. When the solution is applied to inclusion problems, it can be
assumed with sufficient accuracy that the materials are infinitely extended
since the size of the inclusions is relatively small compared to the size of the
macroscopic material samples.
The fundamental equations to be solved for given e*y, B.12), are
c,]kiukjj = cijkit*ki,i- 03-1)
Periodic solutions
Suppose t*j(x) is given in the form of a single wave of amplitude ef,(?), where
? is the wave vector corresponding to the given period of the distribution,
-*), C.2)
kxk.
where i = -J-l and ?• x = i
The solution of C.1) corresponding to this distribution may also be ex-
expressed in the form of a single wave of the same period, that is,
«.¦(х)=йД0«р(/«-х). C.3)

Chap. 1 General theory of eigenstrains
Substituting C.2) and C.3) into C.1), we have
where in the derivation (/?• x) ,= г?, is used. Expression C.4) stands for three
equations (/' = 1, 2, 3) for determining the three unknowns u, for given e"* •
Using the notation
C.5)
we can write C.4) as
2lu-l + К22п2 + К23п3 = X2,
C.6)
Then, w, is obtained as
where N.. are cofactors of the matrix
ч
(К
K2l
к31
11 ^12 ^13
Л22 Л23
к32 к33
C.7)
C-8)
and ?>(?) is the determinant of A"(^). Note that Kki = Ckjil?j?, = Сш^^¦ =
CijkiiliJ = Kik due to the symmetry of the elastic constants CiJkl= CkliJ, and
that Ntj = NJr The explicit expressions for D($) and Ntj(?) are
C.9)
n2KтЪКп\ + €mn3KmlKn2 ) Sij'
where tijk is the permutation tensor.

j. General expressions of elastic fields for given eigenstrain distributions 9
Substituting C.7) into C.3), we have
мДх)= -iCjlmni*mn{Z)mM)D-\!.) exp(i*-x). C.10)
The corresponding strain and stress are obtained from B.2) and B.4) as
e,J(x)=12Cklmni*mn(O^{^ikU) + UiNjka)}D-1(U)^p(H-x) C.11)
and
{(l)exp(^-x)-c*/(x)}, C.12)
where Dl = 1/D. The above result was used by Mura A964) for periodic
distributions of dislocations and by Khachaturyan A967) for a coherent
inclusion of a new phase.
Method of Fourier series and Fourier integrals
The linear theory of elasticity allows for the superposition of solutions. If
t*j(x) is given in the Fourier series form,
<?,(*) = I<7,-@exp(i€-x), C.13)
the corresponding displacement, strain and stress are then obtained as super-
superpositions of the solutions for single waves of the form C.2), namely,
C.14)
where the summations in C.13) and C.14) are taken with respect to ?.
Similarly, if e* is given by the Fourier integral form (see Appendix 3),
e* (*) = Г «?,(«) exp(i{-x) d{, C.15)

10 Chap. 1 General theory of eigenstrains
where
?"*(?) = {2-nY3 f с* (х) exp(-tf-x) dx, C.16)
* — oo
we have
Ui(x) = -if Cj!mni*mntt)Z,N,j@D-l(S) exp(i«-x) d?,
where
/
OO y.00
— со — oo —oo —со
,.00 /-ОО ,.00
/ dx = / dxx I d;
— oo — oo —oo "—oo
When C.16) is substituted into C.17), we have
C.17)
Xexp{i?-(x-x')} d^dx'
"л:/ •'-oo•'-oo
Xexp{i|-(x-x')} d?dx', C.19)

3. General expressions of elastic fields for given eigenstrain distributions 11
xexp{i{-(x-x')}d?dx', C.20)
* — oo — oo
Method of Green's functions
When Green's functions Giy(x — xf) are defined as
C.19) can be written as
/¦00
Сlmni*mn\x )G0 Дх —x ) dx , C.23)
-oo
where %,(х - x') = 3/3x,G0.(x - x') = -3/3x,'C?iy.(x - x'). Sometimes
Green's functions are called the fundamental solutions.
The corresponding expressions for the strain and stress become
/00
С e* (x')\G (x — x') + G (x — x')} dx' C-24)
and
Mura A963) rewrote C.25) in the form
/•00
€sth€tnh^pqmnUkp,qt\X X )tsm\X ) пХ > (j.lO)

12 Chap. 1 General theory of eigenstrains
which will be useful for the dislocation theory given in later sections. It is easy
to prove that C.26) is equivalent to C.25). Since tsthtlnh = 8slSln - 8sn8tl, C.26)
becomes
/00
Cpqmn{GkP,qni*m,-Gkp,qli*mn) их'. C.27)
-oo
In Section 5 it is shown that
CmnpqGpk,qn{x-x')= -8mkS(x-x'), C.28)
where S(x — x') is Dirac's delta function having the property
Г e*ml(x')8(x-x')dx' = e*ml(x); C.29)
— 00
therefore, C.25) follows from C.27).
It is seen from C.28) that Green's function Gpk(x - x') is the displacement
component in the x^-direction at point x when a unit body force in the
x^-direction is applied at point x' in the infinitely extended material. By this
definition of Green's function we can directly derive C.23) from C.1). As was
mentioned in Section 2, the displacement м, in C.1) can be considered as a
displacement caused by the body force - ClVmne* и>/ applied in the x,-direction.
Since Gjj{x — x') is the solution for a unit body force applied in the x -direc-
-direction, the solution for the present problem is the product of Gi and the body
force -CJlmne*mnJ, namely,
G^x-*0W«»./(*')djr'. C.30)
o
Integrating by parts and assuming that the boundary terms vanish, we have
«,.(*)= Г Слтп1*тп{х')^-Си{х-х')&х'. C.31)
For an infinite body it holds that (9/3x;)G,.y(x - x') = -(d/dx^G^x - x');
C.23) is thereby obtained.
Expression C.31) or C.23) is preferable to expression C.30). When с* п is
constant in ?2 and is zero in D-?2, it can be seen that the integrand in C.30)
vanishes except on the boundary of Q.
As will be seen in E.9), GtJ{x-x') has a singularity at x = x' with the
order of | x — x' \ ~x. Thus, the integrals in C.24) and C.25) do not exist in the

3 General expressions of elastic fields for given eigenstrain distributions 13
sense of Riemann integrals. This difficulty can be avoided by writing C.25) in
the form
/ Э r°° \
0,/*)= ~СЦэ^/ jCpim&nixlGkp^x-x^dx' + 'Mx)}- C-32)
Expressions C.25) or C.17) are permissible in the context of generalized
functions (Lighthill 1964).
Expressions C.23), C.25) and their equivalents were developed by Fred-
holm A900). In connection with the solution for dislocations, many papers
have discussed these expressions more extensively: Weingarten A901), Vol-
terra A907), Somigliana A914), Burgers A939), Leibfried A953), Eshelby
A961), Kroner A958), Steketee A958), de Wit A960), Indenbom A966),
Kunin A964), Kosevich A965), Bacon, Barnett, and Scattergood A978), Hirth
and Lothe A982), Teodosiu A982), and Steeds and Willis A979), among
others.
As will be seen in Section 5, Green's functions have been obtained explicitly
only for isotropic and transversely isotropic materials. Therefore, for practical
calculations the usage of Green's functions as seen in C.23) ~ C.25) is limited,
and the use of Fourier integral expressions C.17) is much more convenient.
For this reason the integrands appearing in C.17) are written down in detail.
Isotropic materials
C.33)
e}
where e = ?ktk,
(Л {
n)teJn}, C.34)
= (Л

14 Chap 1 General theory of eigenstrains
Cubic crystals
^2(i+ц')еш22+mi+mi)
C-35)
C.36)
and the other components are obtained by the cyclic permutation of 1, 2, 3,
where
= CU, C.37)
Hexagonal crystals (transversely isotropic)
= (a'T,2 + y«32){ ayij4 + (aj8 + y2 -
= («V + Yl32){(^2 + ^32)(^2 + V^)-Y'Vl32}, C-38)
Nn(i) = (a'f? + a|| + y*i)(y42 + ^f ) - Y%2^2,
^u(€) = Y'2«i«2«l - (« ~ «')«i«2(y42 + )8?2),
JV13(O = (a - «ОУ^Шз - 7'^зИ2 + «^2 + Yl2),
Na($) = (a{? + a'g + У132)(УЧ2 + ^|) ~ У'2Ц%, C-39)
JV23(|) = (a - «')Y'^2?3 - ЧЫА«? + «'И + ^з2),
ЛГ33(^) = («^ + а'Ы + Y€f )(a'{? + «I2 + Y^32) - (a - «'J^,

4 Exercises of general formulae 15
where
a = Cn = C22, a' = C66 = j(Cn - C12),
/3=C33, y'-y = C13=C23, C.40)
For isotropic materials, a = /J = \ + 2ц, у' = Л + /ч, and у = «' = I*-
4. Exercises of general formulae
General formulae are given in Section 3 for the elastic fields associated with
prescribed eigenstrains. This section provides exercises in the usage of these
formulae: the best way to understand general statements is to work out
specific examples.
A straight screw dislocation
Let us consider the case where ?2 in Fig. 1.1 is the half plane (x2 = 0, Xj < 0)
(see Fig. 4.1), and e*3 is prescribed on 12. Other components of e*j are zero.
^..--Positive screw
I dislocation
line
Fig 4 1 A screw dislocation

16 Chap. 1 General theory of eigenstrains
The boundary of Q (the x3-axis) is called a straight positive screw dislocation
when ?*з = ?*2 is given by
Xl), D.1)
where H( — xx) is the Heaviside step function (see Appendix 3),
and 5(*2) is Dirac's delta function. The plastic strain (eigenstrain) D.1) is
caused by a relative slip b on the half plane (x2 = 0, xx < 0) in the ^-direc-
^-direction (Fig. 4.1).
The Fourier transform of D.1) is, from C.16),
+ |2х2 + ^х3)} d^dxjdxj. D.3)
By using
f Н(-хг) exp(-i?lXl) dXl = -\/iix,
D-4)
/ exp(-it3x3)dx3 = 2ir8(Z3),
J — oo
we have
&T-
Formulae C.17) are used as follows: assuming, for simplicity, that the material
is isotropic, C.34) is used in C.17); then, we have
D-6)

4 Exercises of general formulae 17
The following integrals are useful for further calculations; these are ob-
obtained from formulae 444, 633, 415, 632, and 638 • 1 in the Table of Fourier
Integrals by Campbell and Foster A948):
|1d|2 = 277 tan \x2/xx)
Then, D.6) becomes
иъ{х) = (b/2w) tan-1(x2A1) D.8)
which has been obtained by Burgers A939). Other components of м, are zero.
Equation D.8) can also be obtained from C.23), where G,- for isotropic
materials is given in Section 5.
. When D.1) is substituted into C.23), one arrives at
where

18
Chap. 1 General theory of eigenstrains
Now, with the aid of integrals
00 (AfAx'i
2/3
D.10)
v5
D.9) is reduced to D.8). The stress field can be obtained from the displace-
displacement field by the use of B.4).
A straight edge dislocation
Let ?2 be the half plane (x2 = 0, хг< О), and e*i be prescribed by
€*2l(x) = \b8(x2)H(-Xl). D.11)
The x3-axis is then a straight positive edge dislocation (see Fig. 4.2). The
plastic strain t\x is caused by the relative slip * on the half plane (x2 = 0, xl <
0) in the Xj-direction.
Substitution of D.11) into C.16) leads to
\~21
D.12)
Fig 4 2. An edge dislocation

4. Exercises of general formulae 19
For isotropic materials, we have, from C.17) and C.34),
D.13)
This result was obtained by Koehler A941), correcting Taylor's result A934).
Similarly, other components become
«,00-0.
Note that w2(x) in D.14) differs by a constant from that in most text books
on dislocations (e.g. Read 1953, p. 116), because differences in constant values
of displacement components are admissible in the elasticity theory.
The same result can be obtained by the use of Green's functions for
isotropic materials. From C.23) and D.11) we have
Cjl2lbGijtl{x - x') dx'3. D.15)
o
The above expression becomes, from E.10),
D.16)
where x^Xj-x^, x2 = x2, x3 = -X3. By using D.10) the integrations in
D.16) can be done easily, leading to D.13) and D.14).

20 Chap. 1 General theory of eigenstrains
Periodic distribution of cuboidal precipitates
Consider cuboidal regions of inelastic strain (eigenstrain) due to solute segre-
segregation forming cuboidal precipitates. Let the edge length of each be 2a (with
edges along xv x2, and x3), and let these regions be spaced 2L apart in all
three directions. Within each region the eigenstrain will be assumed to have a
constant value, t, and be zero outside this region,
5ое in cuboidal regions
0 outside.
According to Appendix 3, the Fourier series expression for D.17) is
*(х\ = я t Y а exnli—(nx + ax + rx U D 18)
p,q,r= — oo
where p, q, and r are integers and
a , = (w) (pqr) sinl—— sinl—=— I sinl —=— I. D.19)
pqr \ J \ГЧ / \ L I \ L I \ L J V '
Expressions in C.14) are used for the elastic field, where
emnKP, Я, П- mneapqr,
%i='np/L, ?,2 = -nq/L, |3 = 7rr/L,
and the summations are taken with respect to p, q, and r. Namely,
«,.(*)=-i ? CjlmnSmneap^,Nu(i)D-\i)
p,q,r= —oo
For isotropic materials we have
. / pira \ . / qira \ . I гтта
, Л q3X + 2MLc " Sm( —)sm( —)sm( —
D.22)

5 Static Green's functions 21
The other components of the displacement can be obtained by the cyclic
permutation of p, q, r and xv x2, x3. The summations in D.18) and D.22)
exclude p = q = r = 0 in order to have the convergency of series. Since the
term corresponding to/> = <7 = r = 0in D.18) represents a uniform eigenstrain
in the whole space, the term in the displacement field D.22) corresponding to
the uniform eigenstrain can be ignored. Sass, Mura and Cohen A967) calcu-
calculated the diffraction contrast of the cuboidal inclusions by using D.22) and
two-beam dynamical equations. Their electron microscope observation of an
Ni-Ti alloy provides good support of the theory.
5. Static Green's functions
In Section 3 Green's functions GtJ{x — x') have been formally defined by
C.22),
The elastic field quantities for a given е*Длг) are determined by C.23) ~
C.25) if Gu{x - x') are known.
It can easily be shown that G,y satisfy
C,]klGkmJj{x - x') + 8imS(x - x') = 0, E.2)
where S(x — x') is the three-dimensional delta function which is zero every-
everywhere, except at point x = x', and it gives
ff(x')S(x-x')dx'=f(x) E.3)
for any continuous function f(x) where the integration domain contains point
x The proof of E.2) is as follows:
From E.1), we have
Г CljklNknD-^j exp{i*• (x-,')} d|
KikNkmD~^ exp{/«•(*-*')} d|, E.4)

22 Chap. 1 General theory of eigenstrains
where Kik = Cijkl^,. Since Nkm is the cofactor of K, we have
KikNkmD-^Sim. E.5)
On the other hand, Dirac's delta function can be defined (see Appendix 3) as
S(x-x') = S(x1-x[)8(x2-x'2)8(x3-x3)
-(*-x')}d€. E.6)
Thus, E.2) is proved. Equation E.2) is the equation of equilibrium with
respect to a displacement Gkm(x - x') when the body force Xt(x) = SimS(x -
x') is applied. In other words, G, -(x - x') is the x(-component of displacement
at point x when a unit body force in the x^-direction is applied at point x'.
The region occupied by the material is assumed to be infinitely extended.
Since Gtj(x — x') = Gjt(x - x'), the directions xt and Xj in the above state-
statement are interchangeable. It should be noted that Nt (?) and D(?) are
homogeneous polynomials with degrees four and six, respectively. Thus we
have, G{J(x - x') = G,/x' - x).
The integration in E.1) requires exphcit expressions for K, Nt , and D
mentioned in Section 3.
Isotropic materials
By substituting C.33) into E.1), it follows that
, ,oo (X + 2ju)e,,^(\ + ju)ff
Gu(x) - Bw)-3 [ '- \'*\.4 M^'tye4>(,-t-x)d{, E.7)
•' m(X + 2/i)?4
where x' is taken as zero without loss of generality. After the triple integra-
integrations are carried out, we obtain
where x = (х;л:гI/2, or
G(') (J
7
-v)
E8)
E-9)

5. Static Green's functions
23
Fig 5 1 The unit sphere S2 in the J-space Green's function at point x is expressed by a line
integral along S1 which lies on the plane perpendicular to x
where X, fi are the Lame constants (/* = shear modulus), v is Poisson's ratio,
St is the Kronecker delta, and | дс — x' | 2 = (x, — x,')(x, - л:,'). This result was
found by Lord Kelvin A882).
The following expression is frequently used for isotropic materials:
i "mnXi , -j XmXnXi
X5
E.10)
where x = x — x' and | x - x' | = 3c.
The details of the integration of E.7) are as follows: The volume element d?
is defined as
d? = ?2d|dS, E.11)
where ? = (?,?,I/2 and dS is the surface element on the unit sphere S2 in the
?-space, centered at the origin of the coordinates ?, (see Fig. 5.1). By denoting
€ = «, x = xx E.12)
(note that x has been used in E.10) with a different meaning), we can write
E.7) as
— ехр(г?х|-дс)
E.13)

24 Chap. 1 General theory of eigenstrains
When ? in E.7) is replaced by -? as a new variable, we have
exp(-/?-x)d?.
«> /(X + 2ju)?4
Then, similarly,
5,.-(X
E.15)
The last result has been obtained by the transformation of | -> -1. By adding
E.13) and E.15), we have
E-16)
The integration with respect to ? leads to Dirac's delta function. Thus,
Denoting the angle between ? and x by в (see Fig. 5.1), we have
0d0,
)d^
where ф is defined on the plane perpendicular to x as shown in Fig. 5.1. The
starting line for measuring ф can be arbitrary and E.17) becomes
dф
= 1^ (x + W (x^)CC 19)
2x Tsi д(\ + 2ju.)
where Sl is the unit circle on S2 intersected by the plane perpendicular to x.

5. Static Green's functions 25
? located on Sl can be written as
I = ax. cos ф + a 2i sin ф, E.20)
where aXi is the unit vector along Oy, and a2; is the unit vector lying on the
plane of S1 and normal to Oy. Then,
? d<> = -n{aXiaXj + a2;a2;) = wEo - x,.*,.). E.21)
The last result is obtained by the orthogonality of the three unit vectors aXi,
a2i, and xt,
aXiaXj + a2ia2] + xixj = 8tJ E.22)
which is the scalar product of the unit vectors along the xt- and x^-axes.
Green's functions for the two-dimensional plane strain problems can be
obtained by considering the elastic field due to a distributed line force along
the x3-axis,
GtJ{xx-x'x,x2-x'2) = f° GtJ(x-x')&x'3, E.23)
where Gt (x - x') is defined by E.1). For isotropic media,
GtJ(xx - x[, x2 - x'2) = {xJ/R2 - C - Av)&tJ log л}/8»A - v)p E.24)
where R2 = (xx - x[J + (x2 - x'2J.
Green's functions for the plane stress can be derived from those for the
plane strain by replacing E with E(l + 2j»)/A + vJ and v with j»/(l + v),
since in both cases these replacements give the same form of Hooke's law.
* Anisotropic materials
As seen in C.9), iV|. .($) and D(?) are homogeneous polynomials with degrees
four and six, respectively. We can therefore rewrite E.1) in the same way as we
derived E.17) and E.19),
^//.Ш dS(i) E.25)
and
^> E-26)

26
Chap. 1 General theory of eigenstrains
where x = | л: | and the unit sphere S2 and circle S1 are shown in Fig. 5.1. The
line integral expression E.26) has been investigated by Fredholm A900),
Lifshitz and Rozenzweig A947), and Synge A957).
As will be seen in later sections, it is not always necessary to have explicit
expressions of Green's functions. The form E.25) is sometimes more conveni-
convenient than explicit expressions or numerical representations, when it is applied
to problems of inclusions and dislocations. However, much effort has been
devoted to deriving explicit expressions, approximate forms, or numerical
values of Green's functions. Kroner A953) has factorized D(?) for hexagonal
crystals into three polynomials of degree two and obtained an explicit expres-
expression of Green's function for hexagonal crystals. He has also expanded Green's
function for general anisotropic materials into a series of surface harmonic
functions. Further analytical investigations of Green's functions for aniso-
anisotropic materials have been done by Willis A965), Indenbom and Orlov A968),
and Mura and Kinoshita A971), among others. For cubic crystals, no analyti-
analytical expressions of Green's functions are available except for the two-dimen-
two-dimensional case; see, for instance, Eshelby, Read, and Shockley A953) and Fore-
Foreman A955). However, series expressions are given for approximated solutions
of Green's functions, as seen in the work of Mann, Jan, and Seeger A961), Lie
and Koehler A968), and Brass A968).
* Transversely isotropic materials
The presentation in this subsection follows to a large extent the work of Willis
A965). For transversely isotropic materials, Willis has performed the integra-
integration in E.26) by the residue calculation. Expression D(i) in C.38) is written in
the form
E.27)
where
A1 = a'/y, A2=-
Ру
Py
а
P
a]
P
1/2
1/2
E.28)
For transversely isotropic materials the direction of the x3-axis is chosen as an

5 Static Green's functions
27
Fig 5 2 Coordinate systems and a unit circle S1
axis of symmetry, but the other two directions, xx and x2, can be arbitrary.
For a given x we choose the coordinate system shown in Fig. 5.2. The xt- and
x3-axes and vector x lie on the same plane, and the x2-axis is normal to the
plane; that is, the Xj-axis is chosen on the plane containing the x3-axis and
vector x, and the x2-axis is normal to the plane. A new coordinate system т)х,
т/2, and 7J3 is chosen as shown in Fig. 5.2. The ijj- and rj2-axes lie on the plane
containing Sl, and т\г, |х, r}3 and ?3 are on the same plane. The angle between
the т/j- and Xj-axes is denoted by в. Then for a point ? on S1,
^1=-qlcos6, ^2 = TJ, ?3= -¦q1sin6.
Further transformation is introduced. We set f = rj1 + irj2, where
E.29)
E.30)
Then the factor in E.27) is written as
E.31)
with В- = [Ai cos20 + sin20 + .4,)/sin20(l - At).
Now Cauchy's theory of residues is applied to E.26). If f is a root of the

28 Chap. 1 General theory of eigenstrains
equation
f4 + 2fi^2 + l = 0, E.32)
then so are -?, 1/f, and -1/f. Hence, only two roots, a, and -a,, exist
inside the unit circle S1. Let us consider the case where all At are real so that
a\, a\, and a2 are also real.
When the a, (/' = 1, 2, 3) are all distinct and non-zero, the residue calcula-
calculation yields
(«2-a2)(«2-«2)
where
Nu ((a\ - l) cos в, ——, - (a2 + l) sin в j
/8? = 1/0?, PZ = l/a22, Pi = l/aj.
Expressions N^^, ?2, ^3) have been given by C.39).
We now discuss a degenerate case, which occurs when a' = y, so that
A1 = 1. In this case, Ax(il +1^) + |2 = 1 everywhere on the contour of in-
integration and may therefore be omitted from the integrands. Similar residue
calculation leads to
;

5. Static Green's functions 29
where
^ = (cos в, i, -sin в),
-ig1/2cos0, A+52I/2, -(l-S2I/2sin0), E.36)
-53I/2 cos 0, A +53I/2, -A -Я3I/2 sin )
Alternative analytical expressions of Green's functions for transversely
isotropic materials have been obtained by Lifshitz and Rozenzweig A947),
Elliott A948), Kroner A953), Lejcek A969), and Pan and Chou A976). The
result of Pan and Chou is written as follows:
Gn(x) = I {(A', - B'^/R, - vtxl/R)) + 2v,B;{l/R* - x^/R,R
/ = i
+ D(l/R*-x22/R3Rf),
Gu(x) = G2l(x)= I {-(A'^BDv^/R^-lViBlx^/
+ DXlx2/R3R*2,
-(CnA't - Сиу?в;Jх/(С13 + C^R.R*}, (j = 1, 2)
Gn(x)= I {vlAlx1/RiR*-vi{Al + Bl)x1zi/R3i}, E.37)
-(A, + BjvfiC^p2 + Cnxj)/(C13 + C^Ri } ,

30 Chap. 1 General theory of eigenstrains
G22(x)= ? {{A'i-Bl){vt/Rl-vlxl/R]) + Iv^il/R* - xj/R^f)}
i-l
+ D(l/R*-xl/R3Rf),
where
"i = (Си ~ C13)l/2(Cl3 + C13 + 2C44I/2/DC33C44I/2
+ (C13 + C13I/2(C13 - C13 - 2C44I/2/DC33C44I/2,
 = (C13 - C13I/2(C13 + C13 + 2C44I/2/DC33C44I/2
- (C13 + C13I/2(C13 - C13 - 2C44I/2/DC33C44I/2, E.38)
"з =
2, fc, = [C^/vj - С„)/(С13 + C^),
13 = (CnC33I/2, zl = vlx3,
For C13 - C13 - 2C44 # 0,
A[ = B[ = - (С* - C33i
A'2 =B'2 = (См - Сгъу\)/ЪтСъъС^\ - v22)vl E.39)
"iA = ~v2A2 = (C13 + Q)/4wC33C44(i'22 - v\),
Bt=-Ar
For C13 - C13 - 2Q4 = 0,
{
E-40)
^=^2 = 0,
B1 = B2= - (C13 + C44)/16wC11C44.

5. Static Green's functions 31
Pan and Chou A979) have extended their solution to a two-phase material (see
also Sveklo 1969).
* Kroner's formula
Kroner's formula A953) is a series expression of Green's functions for general
anisotropic materials. Since i\fy(?)/D(?) in E.25) and E.26) is a continuous
function on S2, it can be expanded in a series of surface harmonic functions
which uniformly converge on S2, as follows (Hobson, 1931):
, E.41)
n = 0
where
^^f?), « = 0,1,2,..., E.42)
and Pn is the Legendre polynomial. From the orthogonality of the Legendre
polynomials, we can easily derive
E.43)
When E.43) is integrated over S1, we have
E.44)
where ds(i-) = d<|> is a line element of S1.
The addition theorem of the Legendre polynomials is
+ 2 t {(n~mtPnm(i-*)Pnm{? -x) cos т(ф-ф'), E.45)
where P™{z) is the associated Legendre function of degree n and of order m.

32 Chap. I General theory of eigenstrains
When E.45) is integrated over Sl with respect to ? for a fixed value of ?', we
have, since ? • x = 0 and /si cos m(</> — ф') йф = О,
( Pn(i-?)ds(t) = 2«Pn@)Pn(?-x). E.46)
Substituting E.46) into E.44) and using E.43), we have
). E.47)
Therefore, E.26) can be written as
±? E.48)
which is an expression of Kroner's formula A953) modified by Mura and
Kinoshita A971). The integer n in E.48) takes only even numbers, since
Р„@) = О for odd numbers of n. Un(x) is calculated from
E-49)
The uniform convergency of E.48) has been proved by Mura and Kinoshita
A971). They also showed that
E-50)
where Л is the Laplacian. The above expression is continuously differentiable.
* Derivatives of Green's functions
As seen in Eqs. C.23) ~ C.25), derivatives of Green's function appear in these
formulae rather than Green's function itself. The plane of the unit circle S1 in
E.26), however, is orthogonal to the direction of x and therefore depends on
x. Therefore, the derivatives of Gtj need special consideration. The following
argument follows the work done by Barnett A972) and Willis A975). Differen-
Differentiating E.25) with respect to xk leads to
^ E.51)

5 Static Green's functions 33
where 8' is the derivative of 5 with respect to the argument. Since
f dS(€)=<? йфГ d(i-x) E.52)
Js2 zsl J-i
and 8(xe-x)ikNiJ(i)D-1 = 0 at ?x = l and -1 F = 0, 77), integration by
parts of E.51) yields
2fs[u] E.53)
and therefore
^ [] E.54)
On 51 we have (see Fig. 5.1)
d(i-x) = x d(i-x) = -x sin в йв = -х йв,
-хк, E.55)
where Klm has been defined in C.5). On the other hand, differentiating
KlmNmjD~1 = 8tj with respect to в, we obtain
E.56)
and
{
= NaNjmD-2Clpmq{xplq + ipxq) E.57)
since NnD~lKlm = 8im. Thus, E.54) is written as
E-58)

34 Chap. 1 General theory ofeigenstrains
Higher derivatives can be obtained in a similar manner.
Differentiating E.25) with respect to x^ and x,, we have
, 4f mHjAA^H) (?) E-59)
S J -1
Integrating twice by parts, we arrive at
which is finally written as
-2{(xk?,
) Csatb (xjb
E.61)
Two-dimensional Green's function
Green's function for two-dimensional problems has been given by E.24) for
isotropic materials. For anisotropic materials, the two-dimensional Green's
function has been developed by Eshelby, Read and Shockley A953), Stroh
A962), and Bullough and Bilby A954). It is particularly convenient to use
Green's function for generalized plane problems (mixture of plane strain and
antiplane strain).
Green's function Gtj{x — x') for generalized plane problems can be ob-
obtained as the displacement иг{х) when a unit line force F} acting at x' parallel
to the x3-axis is applied. The equations of equilibrium for ut are
<W«*pe/J = 0, i, k = 1,2,3; a, 0 = 1,2, E.62)

5 Static Green's functions 35
for all points except x'. Hooke's law is
o,j = Cijklekl, E.63)
and the strain and displacement relation is
eu=j(«ij + Uj,,)- E.64)
The solution is given by an arbitrary function of a linear combination of the
variables xx and x2 and can be written as
uk = Akf(z), E.65)
where
z = x1+px2. E.66)
Substituting E.65) into E.62), we have
ф,кАк = 0 E.67)
with
tk = Cnkl +p(Cilk2 + Ci2kl) +p2Ci2k2. E.68)
Equation E.67) has a nontrivial solution for the vector Ak if and only if the
determinant of the coefficients equals zero,
1Ф,-*1=0. E.69)
This determinant is a six-order polynomial in p. Its complex roots occur in
conjugate pairs. For a given root p(/), the corresponding values of Ak are
determined from E.67). However, the displacements must be real; this condi-
condition requires that the imaginary parts of corresponding pairs of solutions
cancel out, and we need take only three roots pm, /»B< and /»C), no two of
which are complex conjugates. The displacement equation can be written more
explicitly as
E.70)

36 Chap. 1 General theory of eigenstrains
where z(n = x1+p(/)x2, and Re stands for the real part of the complex
function in the braces. The stress components are
( 3 )
a,, = Re E [CiJkl +pafijk2\ AwkfmUu)]}- E-71)
If /(/)[z(/)] is not a single-valued function, the singular line represents a
Volterra dislocation. On the other hand, if /(/)[z(/)] is a single-valued function
but the resultant force around the singular line is not zero, the singular line
then represents a line force per unit length, Fr The conditions necessary for
obtaining Green's function are
,/i,d* = *;. E.72)
and
[dukds = 0, E.73)
where с is a closed curve around point x', and n, is a unit vector normal to с
Condition E.72) can be written as
*,. = F,, E.74)
where ф( is defined by
- E-75)
From E.71), E.68) and E.75), we have
? [Cnkl+pwC,2k2]A(l)kf(l)[z{l)]\. E.76)
Eshelby, Read, and Shockley A953) have shown that
/(/>[*</)] = (l/±2*n)A/) MzO)-z'(o] E-77)
with
*</) = *{+/\/)*i> E-78)

5. Static Green's functions 37
where the sign of 2w/ is taken to be the same as the sign of the imaginary part
of p{ly Then, conditions E.74) and E.73) become
Rej ? [C/2*i + Paftiki] Л/Л,} = F, E-79)
and
Equations E.79) and E.80) represent a system of six simultaneous linear
equations for six unknowns D(l).
The properties of the coefficients A(j,k were studied in detail by Stroh
A962). D(/) can always be expressed as a linear combination of Ft,
D(l) = dU)mFm. E.81)
Now let us define a complex variable X= xx + ix2 (with its conjugate
X= x1 - ix2). It follows that
() [wpwX]. E.82)
The displacement then is
з
/-1
-(X' + X'-ipwX' + iPil)X')]\. E.83)
Finally, Green's function is obtained from E.83) as
'- ip(l) X' + ip(l) X')]\. E.84)

,o Chap 1 General theory of eigenstrains
6. Inclusions and inhomogeneities
Inclusions
An inclusion is defined as a sub-domain & in domain D, where eigenstrain
e* (x) is given in fl and is zero in D-?2 The elastic moduli in fi and I>-S2 are
assumed to be the same. The remaining domain D-Q is called the matrix.
From the result in Section 3, the elastic field due to the inclusion can be
written as
nC»(x'){GlkJj(x-x') + GjkM(x~x')} dx', F.1)
Cpqmne*mn{x')Gkptql{x- x') dx' + ^kl(
or
— 3 f С
¦'-оо-'й J m" m"
Хехр{/?-(х-х')} d|dx',
-оо-'й
F.2)
-oo •'fl
wberee*/(x) = 0forxe2)-O.
шее «*Дх) is discontinuous on the boundary of domain J2, some quanti-

6. Inclusions and inhomogeneities 39
ties may also be discontinuous on that boundary. However, the displacement
and the interfacial traction across the boundary must be continuous; that is,
F.3)
[ a,7] Hj = { a,j (out) - a,7 (in)} и, = 0,
where и. is the outward unit normal to the boundary of fl. When an inclusion
or inhomogeneity can slide on the interfacial surface, the first condition in
F.3) does not hold. This case will be discussed in Section 51. As discussed by
Hill A961) and by Walpole A967), the displacement gradient (distortion) u4
is continuous inside п and D-Q but discontinuous at the interface between the
two domains. The jump across the interface can be written as
[«,,>] s ",,>ut) - ",,,(in) = \nj. F.4)
Л, is the proportionality constant (the magnitude of the jump) to be de-
determined. The above equation is designed so that the condition [и,. ] dXj = 0
for dx along the boundary of J2 can be satisfied. This condition stems from
the continuity of displacement along the boundary (и^ dXj = 0 for dx along
the boundary).
Since
°U=CUki(uk.i-et,), F-5)
the second equation in F.3) is written as
Cijkl{[ukJ] -[??,]}«,¦ = 0, F.6)
where [t*kl] = e*,(out) - €*,(in), e*/(out) = 0, and e*,(in) = c?,. Substitution of
F.4) into F.6) gives
cijkixkninj= -Cijkrf/nj- F-7)
Equation F.7) is a system of equations to determine X for given n and e*.
Since the system has the same form as C.4), we obtain
K= -CJkm**mnnkNtj{n)/D(n), F.8)

40 Chap. 1 General theory of eigenstrains
where И^{п) and D(n) are the cofactor and determinant of the matrix K(n),
namely,
D(n)=emnlKmlKn2Kl3,
F.9)
) \
where Kik = CiJklnJnl. Substituting F.8) into F.4), we arnve at
K,] = -Clkmne*mnnknjNu(n)/D(n) F.10)
and, therefore,
[au] = a,., (out) - al7(in) = Cijkl{[ukJ] - [e*,]}
= C,jkl{-Cpqmne*mnnqn!Nkp(n)/D(n)+e*kl}. F.11)
Equation F.11) is useful for evaluating the stress just outside the inclusion
when the stress just inside the inclusion is known. As shown in later chapters,
eigenstresses inside ellipsoidal inclusions are usually easy to obtain. This
interesting observation, relation F.11), was first made by Goodier A937) in his
paper on thermal stresses.
Stress concentration factors of ellipsoidal inhomogeneities are easily ob-
obtained from F.11), as seen in Section 31. The integrations in F.1) and F.2) are
performed in Chapters 2 and 3 for ellipsoidal inclusions.
When the eigenstrain e* n is uniform in Q, and п has an arbitrary shape, it
is sometimes convenient to rewrite F.1), by integration by parts. Then, the
first equation in F.1) becomes
»,(*) =f Cj^l&jix-x'fr, US, F.12)
where | й | is the boundary of ?2 and dS is its surface element.
Inhomogeneities
When a subdomain of a material has elastic moduli different from those of the
remainder (matrix), the subdivision is called an inhomogeneity. A uniform
applied stress at infinity is not uniform in the neighbourhood of the inhomo-
inhomogeneity. This stress disturbance due to the inhomogeneity can be simulated by

6. Inclusions and inhomogeneities 41
an eigenstress field caused by an inclusion when the eigenstrain is chosen
properly, as seen in Section 22.
For plane strain or plane stress inhomogeneity problems (composite prob-
problems), the complex potential method of Muskhelishvili A953) is more effective
than the equivalent inclusion method (see Sherman A959), Hardiman A954),
Sendeckyj A970), and Theocaris and Ioakimidis A977), among others).
Numerical examples of the application of the complex potential method are
given for multiple circular inclusion problems by Ют and Ukadgaonker
A971) and Yu and Sendeckyj A974). Sendeckyj A970, 1971) also gives the
exact solution of the antiplane longitudinal shear deformation of composites.
In antiplane problems, the nonzero displacement and stress components are
given in terms of one analytic function (see Milne-Thomson, 1962). Recently,
Wheeler A985) has considered the problem of minimizing stress concentration
at a rigid inclusion.
Equations similar to F.3) and F.4) hold for an inhomogeneity under an
applied stress. Let us assume that i2 is an inhomogeneity with the elastic
moduli C*kl. The displacement and the interfacial traction across the boundary
| Q | must be continuous and the relations F.3) and F.4) hold.
Since
F.12.1)
%^) = C*jklukl{m),
the second equation in F.3) is written as
CljklukJ(out)nj = C*klukl(m)nj. F.12.2)
The relation F.4),
u,,y(out) = Л,.и, + и,/in), F.12.3)
is substituted into F.12.2). Then, we have the equation to determine the
unknown vector X,
c,jkiXkninj = Acijkiuk,i(in)nj> F.12.4)

42 Chap. I General theory of eigenstrains
According to the method for obtaining C.7) from C.4), we have
\, = XkN,k{n)/D{n), F.12.5)
where
Xk= ACklmnum Jin)n,. F.12.6)
Substituting F.12.5) into F.12.3), we have
и,., (out) = и,,, (in) + Nik{n)njn, ACk!mnumn(in)/D(n). F.12.7)
Consequently, the first equation in F.12.1) is written as
a,,(out) = Cljkl[ukJ(in) + Nkp(n)ninq ACpqmnumJin)/D(n)] F.12.8)
or
ai;(out) = Cljkl\stlah + Nkp(n)Nlnq ACpqmnS*nah/D(n)]aah(in) F.12.9)
where S*k/ is the elastic compliance and
\{Ui Jm) + UjJin)} = S*klakl(m). F.12.10)
The above result F.12.9) has been obtained by Hill A961) and applied to a
thin-coated inhomogeneity by Walpole A978). Numerical calculations for a
coated inhomogeneity have been obtained by Mikata and Taya A985) by the
use of Papkovich-Neuber functions.
It must be emphasized that the equations F.12.7) and F.12.9) hold for any
shape of inhomogeneity.
* Effect of isotropic elastic moduli on stress
Suppose that a composite body consisting of two isotropic elastic phases is
loaded by prescribed surface traction AF, where F is a fixed vector and A is
the load parameter. According to the linear theory, and when proportional
loading is involved, the induced stress is proportional to the magnitude of the
applied tractions. It then follows from dimensional analysis (e.g., Langhaar
1951) that in three-dimensional cases, the stress at any point in the composite
is a function of three parameters involving the four elastic moduli. It is

в Inclusions and inhomogeneities 43
obvious that a^/X, \/ц1, p2/lxi> vv and vi are dimensionless and that otJ/\
can be expressed as
where fix and v1 are the shear modulus and Poisson's ratio of the matrix and
fi2, v2 are the corresponding constants of the inhomogeneity. Since atJ is a
linear function of X, F.13) must have the form
aiJA=f(ii2/ii1,v1,v2). F.14)
Therefore, a^/X is a function of three parameters H2/H1, vx and v2, or three
combinations of them.
An important reduction in the dependence of stress on the elastic moduli
takes place when the geometry and loading of the composite body are such
that it is in a state of either plane strain or plane stress. If the composite body
is multiply connected, and if the vector sums of tractions on holes vanish for
every individual hole, Dundurs A967) has shown that, under these conditions,
the stress depends on only two combinations of elastic moduli. In a sense this
result is the counterpart of the theorem by Michell A899) establishing the
circumstances under which stress is independent of elastic constants in a
homogeneous material. Choice of these two combinations (parameters) of
elastic moduli is not unique. Dundurs A969, 1970) has proposed the following
two parameters (Dundurs' constants):
1)Г+к2+1 '
F.15)
(к1 + 1)Г+к2 + 1 '
where к = 3 - 4v for plane strain and к = C - v)/{\ + v) for plane stress, v
being Poisson's ratio. The subscripts 1 and 2 are used on the elastic moduli to
distinguish between the two phases, and Г = /г2//*1, where ju is the shear
modulus. The a,/?-plane provides a convenient means for classifying com-
composite materials regarding their physical behavior and for exhibiting such
results as stress concentration factors that depend on the elastic moduli.
Because of the physical limits Г > 0 and 1 < к < 3, the admissible values of a
and /} are restricted to a bounded region or, more specifically, a parallelogram
in the a,/?-plane, as shown in Fig. 6.1. Identical materials are represented by
tt = /? = 0, and the case of equal shear moduli, or Г=\, corresponds to the

44
Chap. 1 General theory of eigenstrains
(I, '/2)
-Г-ОЭ
Tig 6 1 The Dundurs constants diagram
straight line a = fi. Representation of stress concentration factors for a com-
composite in terms of a and /? is found in the paper by Gerstner and Dundurs
A969).
7. Dislocations
The mathematical theory of dislocations in elastic continua was first sys-
systematically studied by Volterra A907) and Weingarten A901). Taylor A934),
Orowan A934) and Polanyi A934) independently used dislocations to explain
plastic deformation of single crystals. They considered dislocations as imper-
imperfections in crystals and explained why observed yield stresses of crystals are
much lower than the theoretical values calculated from atomic theory on the
basis of perfect-lattice state. The existence of dislocations was first confirmed
directly by Hirsch, Howie, and Whelan A960) by transmission electron
microscopy (indirect observations by etch pits, etc., had been done earlier—see
Hirth and Lothe 1982). The direct observation of dislocations has been
extensively made (see Amelinckx 1964), and the dislocation theory has been
developed by many scientists to explain not only the mechanical, but also the
optical and electromagnetic properties of crystals (see Nabarro 1967, 1979).

7 Dislocations 45
-L
Fig 7 1 Dislocation Л and the Burgers circuit с
As already explained by Figs. 4.1 and 4.2 in Section 4, the dislocation (line)
is defined as a part of the boundary of a slip plane, which is embedded in a
material. The part of the slip boundary which is exposed on the surface of the
material is not called the dislocation line. To define the direction of a
dislocation line in a more precise manner, consider a slip plane S inside a
material as shown in Fig. 7.1. It is assumed that, for instance, the upper plane
(denoted by S+) slips by b relative to the lower plane (denoted by S~). In
order to specify this configuration of the slip, we define the direction v of the
dislocation line L in the following manner: Go around a linking circuit с (the
Burgers circuit) in the direction of rotation of a right-handed screw advancing
along the direction of the dislocation. The surface S+ that is displaced by b
(the Burgers vector) relative to the other surface 5' is designated as the
surface on which the end point of the Burgers circuit is located; the circuit
does not cross the boundary between S+ and S~, as shown in Fig. 7.1.
For a crystal, the Burgers vector of a dislocation is usually a lattice vector.
Such a dislocation is called a perfect dislocation. A dislocation which has the
Burgers vector not equal to a lattice vector is called an imperfect or a partial
dislocation. The slip deformation in crystal plasticity is caused by the move-
movement and generation of dislocations, and is generally irreversible due to the
atomic potential barrier. Due to the singularity along the dislocation line, an
elastic field is introduced in the material. This dislocation state of the material
is simulated by an'eigenstress field caused by eigenstrain ef as discussed in
Section 4.
Vol'terra and Mura formulas
According to Kroner A958), the displacement gradient м, 7 (total distortion) is
assumed to consist of elastic distortion j8y, and plastic distortion {}*,
"<., =/*,, + /** ¦ G.1)

46 Chap. 1 General theory of eigenstrains
The (total) strain ?,y, elastic strain e,y, and eigenstrain e*, defined in B.1) ~ B.4)
are
G-2)
Since ^^ is caused by the slip bt of plane S+ whose normal vector toward S
is n , we can write
)8;(x)=-6,11/E-*) G.3)
where S(S - x) is the one-dimensional Dirac delta function in the normal
direction of 5", being unbounded when x is on 5 and zero otherwise. As in
D.1) and D.11) we write
JJ Jx). G.4)
When this expression is substituted into F.1), and since
J8(S-x')dx'=fdS, G.5)
S
then we have the Volterra formula A907),
тОиЛх-х')пп dS(x'). G.6)
Differentiating G.6) we have
CklmnbmGikJj(x- x')nn&S{x'). G.7)
G.8)
j
s
The elastic distortion is obtained from G.1) as

7. Dislocations 47
By integrating G.8), the Mura formula A963) is obtained as
?,(*) = jejnhCpqmnGiPiq{x- x')bmvh dl(x'), G.9)
where v is the direction of the dislocation line L and d/ is the dislocation line
element. The equivalence of G.8) and G.9) is easily proved by the use of
Stokes' theorem,
ffvhdl= f4lhfjnkdS. G.10)
Jl js
Applying G.10) to G.9) we have
?,,(*) = -jiklhijnhCpqmnGip,ql{x-x>)bmnk dS(x'). G.11)
The minus sign comes from (Э/Здг,')^^ = — Gipql.
Since eklhtJnh = 8kJ8ln - 8kn8h,
mnGiPtqnbmnj - CpqmnGipqjbmnn) dS. G.12)
From E.2), CpqmnGipqn(x-x')= -8m,8(x-x'); therefore the first term in
G.12) is written as
[S(x- *')M, dS(x') = b^SiS - x). G.13)
Js
The two Dirac delta functions in G.13) are different. 8(x - x') is the three-di-
three-dimensional delta function defined by E.6), and 8(S — x) is the one-dimen-
one-dimensional delta function in the direction normal to S. Equation fs8(x — x')
dS(*') = 8(S— x) can be regarded as the defining equation for 8(S ~ x).
Now it can be seen that G.12) equals G.8). Systematic derivations of G.9) are
presented by Willis A967) and Teodosiu A970).
The stress components are
°ij = Ci,kiPik> G-14)
or, from G.9),
'nhCpqmnGkp,q(X ~ X')bmVh d/(*')- G-15)

48 Chap. I General theory of eigenstrains
The fact that the elastic distortion and stress fields can be expressed by line
integrals along the dislocation line proves that these fields depend only on the
boundary of S. On the other hand, the displacement field cannot be expressed
by such a line integral, which means that the displacement field cannot be
determined uniquely by only L. It can, however, be determined by knowing S,
as shown in G.6). If a history of the origin of dislocation (i.e., how it has been
created) is given, the displacement field can also be expressed by a line
integral along the dislocation line (see Section 38). Giving S and b for a
dislocation is equivalent to giving a history of creation of the dislocation line.
The Fourier integral expression for G.9) is
J' B-rrK l -°° ;" Pimn q m
Xexp{/? •(*-*')} d? G.16)
which will be useful for further calculations (see Section 38).
It should be mentioned that ut Ax) = jS-Дл:) when point x is not on S, and
therefore G.9) or G.16) can be regarded as expressions for и;у(лг) at such a
point.
Equation G.6) holds even when b is a function of x. Such a b is called the
Somigliana dislocation. The Somigliana dislocation is useful for describing
some interfacial defects as seen in the papers by Asaro A975), Bonnet,
Marcon, and Ati A985), and Mura and Taya A985) and in Section 51.
* The Indenbom and Orlov formula
The line integral expression for Pjt(x) or а,7(дс) as shown in G.9) or G.15) is
not unique. An alternate expression was obtained by Indenbom and Orlov
A968),
PA*) = iIvpv«$x \x №*'') d/(*')' G17)
where
A*-*) = t^tj Г h <ь Г ijnh
Btt) /-«> ¦'-«>
G.18)

49
\
\
\
Fig 7 2 Explanation of notations appearing in the Indenbom and Orlov formula
and v is the unit tangent to L at x'. The unit vector along лг'л: is denoted by t.
It can be seen from G.16) that /?,,-( у; t) is the elastic distortion at the end
point of vector v caused by a straight, infinitely long dislocation connecting x
and x' points (Fig. 7.2).
The interesting point of expression G.17) is that the elastic stress field of
curved dislocation line L can be expressed in terms of the stress field of a
straight infinite dislocation. However, the appearance of the second derivatives
in the integrand makes G.17) less attractive unless it is modified as done by
Asaro and Barnett A976). This is discussed in Section 36. The equivalence of
G.17) and G.9) has been proved by Asaro and Barnett A976). Further
modifications of G.17) have been developed by Lothe A960), Brown A967),
Asaro et al. A973), Bacon, Barnett and Scattergood A978) among others.
* Disclinations
Volterra A907) considered two types of dislocations: one is translational and
the other one is rotational. The translational dislocation is the dislocation
shown by Fig. 7.1. The rotational dislocation is called disclination and is
shown in Fig. 7.3. The name "disclination" was coined by Frank A958) in his
study of cholesteric liquid crystals. Liquid crystal disclinations have been
observed as twisting discontinuities, allowing discrete jumps of one half-pitch
of the helicoidal texture (e.g., Orsay Liquid Crystal Group 1969, Friedel and
Kleman 1969, Robinson, Ward and Beevers 1958, Cladis, Kleman and Pieran-
ski 1971, Williams 1975). Trauble and Essmann A968) have made direct
observations of disclinations in the lattice of flux lines in type II superconduc-
superconductors and have shown that disclinations cause strong elastic distortions and
lattice bendings. Nabarro A967, 1969) and Harris A969) have extended the

50 Chap. 1 General theory of eigenstrains
Fig 7 3 Disclination L
concept of the disclination to surface defects and suggested possible applica-
applications to biological structures such as some plasma membranes, virus capsids
and insect muscle (see also Harris and Scriven 1970, Harris and Thomas 1975,
Chou and Ichikawa 1979, and Ichikawa and Chou 1979). Disclinations are
also created by multiple twinning and have been observed in FCC crystals
(Smith and Bechtoldt 1968). Li and Gilman A970) have shown that disclina-
disclinations are caused in polymers by chain kinking and twisting of molecules and
that they play an important role in plastic deformation, internal friction, and
glass transition. Recent developments of disclinations are seen in the mono-
monographs by Kleman A980, 1983) and Bouligand A980).
The disclination (line) L in Fig. 7.3 is created by twisting surface S+
against surface S~ by rotation angle и at point дг°. Surfaces S+ and 5"" are
defined by the direction v of the disclination line and the advancing rule of
the right-handed screw, as employed for the definition of dislocations (see Fig.
7.1). Surface S+ will be denoted by S. When vector и is normal to S, the
disclination is of a twist type, and it is of a wedge type when u> is on S.
The elastic field caused by a disclination is simulated by the eigenstress
field caused by a suitable eigenstrain e* distributed on S. Since the jump in
the displacement components across S is
the plastic distortion on S is defined by
{x) G.20)
which is similar to the definition G.3). The corresponding eigenstrain is
defined by G.2). When G.20) is substituted into F.1), where п = S, we have
') G.21)

7. Dislocations 51
and
lmntmpfy(x'q-x°q)GikylJ(x- x')nn dS(x'). G.22)
ui j~ P?i= fyi cannot be fully expressed by a line integral along L as done in
G.9); however, the symmetric part, /So/) = iC/S,, + Ay)> can be expressed by
the line integral,
nl,Gik<l(x- x'))sCklmnempqo,p(xq - x°q)vn dl(x')
nHG*k,M - x'))'CklmjinpJ*hPp dl(x'), G.23)
where
3Г -x)di, G.24)
and [ Y is the symmetric part of [ ] with respect to /, j.
Mura A972) found that
- /*jnhCklmnGik,l(X ~ *')«««»«*"* dl(x')
+ fi;iia(x)+eJih4>;h(x), G.25)
where
<t>:h(x)=-u>hna8(S-x). G.26)
Ф*л is called the plastic rotation. Contrary to the dislocation case, м, • cannot
be expressed by the line integral along L. One more derivative of м, is
required. Quantities expressed by line integrals along L can be called state
quantities since in a dislocated state of crystals the only measurable (visible)
line imperfections are lines L of dislocations or disclinations. It can be said
from G.23) and G.25) that P{Ji) and uija - 0Дв - iJih^*h are state quantities.

52 Chap 1 General theory of eigenstrains
In the dislocation case, м, - /?у* is a state quantity. A derivation similar to
G.25) is presented by Kossecka and deWit A977).
Formula G.25) is useful for the derivation of the dislocation density tensor,
asi. Since ut ja = ui aj, it holds that
e,ja«ija = 0- G-27)
When G.25) is substituted into G.27), we have
klmaGik,lfmp4Up{X'q ~ Xq) dx's ~ j CklmSGik,l<f-mPq">P{X'q ~ X°q) dK
~ f Ckln,aGik,l*mabUs dx'b+ \CklmsGik^mab^a dx'b
JL JL
+ e,jaPj*.a + <sjfjib*:k = O> G-28)
where dx's = vsdl. The first term becomes -jL8mi8(x- x')(.mpqwp(x'q- x°q)
dx's, where the property of Green's function is used. The second term becomes
-\LCkimsGik^mPqupbqa dx'a by integrating by parts, and it cancels out with
the fourth term. The third term is zero since Cklmcf.mab = 0. If we define
f8(x-x')dl(x')=8(L-x) G.29)
as in G.13), equation G.28) can be written as
-<W,K - **)*№ ~X)+ *,jaVi.a + t,j*JiH*tk = 0 G-30)
or
<*s, = -<saA*,a ~ tsajtjihtiH, G-31)
where
( ) *)¦ G-32)
Equation G.31) states that the disclination line can be expressed in terms of
the plastic distortion and rotation. Suppose that a crystal is distorted by many
disclination lines. The smearing-out of the density of disclination lines may be
expressed in the same form as G.31) with continuous functions of tensors, asj,

8. Dynamic solutions 53
/?*, and 4>*h. asi is called the dislocation density tensor. The dislocation theory
can be included in the theory of disclinations because — €ipqwpxaq in G.19) is a
constant and can be replaced by the Burgers vector bt. The dislocation theory
is a special case where ф*А = О. Then asi becomes Nye's A953) dislocation
density tensor, and we have
The above equation was first obtained by Kroner A955, 1956). In Section 37,
G.33) is derived by a different method.
The divergence of G.31) leads to
a ¦ = -e- JBh, G.34)
SI ,J IJn Jfl' v '
where
в,= -с Ф*л • G.35)
Anthony et al. A968) first found G.34), and they defined #,7 as the disclina-
tion density tensor. In dislocation theory the relation ahi h = 0 is well estab-
established. It has the form of a continuity equation and has been interpreted to
mean that dislocations cannot end inside the medium. DeWit A969, 1971)
pointed out that G.34) has the form of a continuity equation with a source or
sink and that dislocations can start or end on disclinations (see also Das et al.
1973, Das, Marcinkowski, and Armstrong 1973, and Marukawa 1974). The
recent development of generalized continua (e.g., Kroner 1968) has stimulated
the study of moments, transport law of a motor (Schaefer 1968), the curvature
tensor of Cosserat continua, and the disclination density tensor (e.g., deWit
1972, 1973, Anthony 1970, Kroner and Anthony 1975, and Gunther 1972).
Elastic stress fields and strain energies of disclinations have been investigated
by Huang and Mura A970, 1972), Chou A971), Liu and Li A971), Kuo and
Mura A972, 1973), Kuo, Mura, and Dundurs A973), and Minagawa A977).
8. Dynamic solutions
The eigenstrain problems discussed in Section 3 are easily extended to
dynamic problems.
Consider an elastic homogeneous material which is infinitely extended. The

54
Chap. 1 General theory of eigenstrains
problem is to find the elastic field when prescribed eigenstrains e* are given as
functions of space and time, c* is given by the Fourier integrals as
€"*(€, «) exp{i(€-x-«0}
(8.1)
The equations of motion are
where p is the density of the material. The stress-strain relations are
aij=cijki(uk,i-e*i)-
We assume the solution of (8.2) to be
u,(x, 0= (Г ",(l, <*)exp{i(?-x-wt)} d^dw.
^ •'-oo
Substitution of these equations into (8.2) leads to
By using the notations
ui is obtained from (8.5) as
(8-2)
(8-3)
(8-4)
(8.5)
(8.6)
(8.7)
where Ж, and Z) are the cofactor and the determinant of the matrix
^12 -^13
К21 К22 f
Л31 Л32
К
23
(8-8)

8. Dynamic solutions 55
Now the solution can be written as
u,(x, t)=-ff iCjlmni*mn(L «)«A,(€, ")?"U «)
Xexp{/(?¦*-и*)} d?d«. (8.9)
If we define
G,j{x, t) = {2m)-4r NtJ{i,a)D-l{t u) exp{/(* ¦*-«/)} d* d<o,
•'•'-oo
(8.10)
(8.9) becomes
«,.(*, r)= -/Г W-(*'' *')%/(*-*', '-Odx'dr', (8.11)
where the second equation in (8.1) has been used. Gu(x, t) defined by (8.10) is
called the dynamic Green's function. Willis A965) used (8.11) for a body
which spontaneously tends to undergo an eigenstrain t*mn.
Uniformly moving edge dislocation
The plastic distortion of a uniformly moving edge dislocation with velocity vx
in the Xj-direction is described in a form similar to D.11),
&*(*, t) = b18(x2)H(olt-x1), (8.12)
where H is the Heaviside step function and «J2 = ?*i = i^*i- The Fourier
transform of (8.12) becomes, from (8.1),
fa. = -Mtt3)*tti«i + «)/B^J^ (8.13)
and €*j = e*2 = jP*v Assume that the material is isotropic. Then, (8.9) be-
becomes, by use of (8.13),
u(x л= (Г
Д ' ^ JJ-
tMti, ti,о; -^i) +
l- Vlt) + $2x2}} d^ d^2. (8.14)

56
Therefore
Chap. 1 General theory of eigenstrains
^ d|2,
"l f f
™л2 J-rr.J-,
2c?
-1
-1
(8.15)
|x d|2,
«3 = 0,
where cx = (ju./pI/2 and c2 = {(X +2ju)/p}1/2 are the shear and the dilata-
tional wave velocities. The integrations in (8.15) are easily performed by use of
D.7). Then,
¦nv\
¦7TV,
\/2c\
\-v\/2c\
log
+
(8.16)
1/2
The above result was first obtained by Eshelby A949).

* 9. Dynamic Green's functions 57
Uniformly moving screw dislocation
The plastic distortion of a uniformly moving screw dislocation with velocity vx
in the Xj-direction is
j8?(x, t) = b,8{x2)H{v1t-xl) (8.17)
where e?3 = e%2 = \Р*Ъ. In the same way the reader can easily obtain
x A - v2/c2Y/2
which is the well-known solution found by Frank A949).
* 9. Dynamic Green's functions
The dynamic Green's function defined by (8.10) is also defined as the solution
of the equation of motion
CiJklGkmJj(x, t) + 8im8(x)8{t) = pGim(x, t). (9.1)
Gjjix, t) is the elastic displacement component in the x,-direction at point x
and time /, produced by a unit impulsive force applied in the jcy-direction at
point д: = 0 and time t = 0. G,- • is a symmetric tensor, and therefore the x,- and
^-directions in the above explanation can be interchanged. The solution of
(9.1) is
GtJ(x, t) = B«уЧГ NU(S, co)/)-1^, «)exp{i(*-x-«0} d* dw>
' — oo
(9.2)
where N(J and D are the cofactor and the determinant of the matrix as defined
by (8.8).
It is easy to show that (9.2) is the solution of (9.1). Write
(€-х-«0}<1«<1« (9.3)

58 Chap. 1 General theory of eigenstrains
and substitute it into (9.1). Then,
(Klk-po>%k)Gkm = 8im, (9.4)
where
8(x)8(t) = Bir)~A ff" exp{i(t-x-at)}dSda (9.5)
' * — со
is used. The solution of (9.4) is
G^U, <o)=^(?, <o)/?(?, ")• (9.6)
The explicit expressions for D and Ntj are
o) = - (ры2K + (рш2J^ - рш2В + С,
() ()
(9.7)
NiJU,w)=KimKmJ+(pU2-A)Kij+{(Po>2J-Pu2A+B}8ij,
where
emn2Km3Knl + <mn3KmlKn2, (9.8)
The roots of D(?, w) = 0 with respect to pw2 are denoted by pw2, pw\ and
j. Then we can write
D(S, a) = (pw2 - pco2)(pco2 - pco2)(pco2 - po>2), (9.9)
and therefore,
A = pw2 + pwj + P,
B = p2w2co2 + p2w2w2 + p2w2w2, (9.10)
C = p3w2w2w2.

* 9. Dynamic Green's functions 59
It should be noted that Z>(?, to) becomes D(i-) when со = О and D(?) is
positive due to (9.10). G,- can then be written as
3 II
G,jti,o)= L 2 J 2, (9.П)
where
(9.12)
and ф;ф., ф,^ are obtained by cyclic permutation of A, 2, 3). ф; are eigenvec-
eigenvectors of (8.8), and ры„ are the corresponding eigenvalues,
v = l
It should be noted that фД. are homogeneous functions of | of degree 0, since
pu>l are homogeneous functions of ? of degree 2, that is,
(9.14)
« = «.
The integral (9.3) can be written as
X f Ф,A)Ф,A") exp(i#~- x) dS(€) (9.15)
where | = || and 52 is the unit sphere \?\ = 1. To integrate with respect to со,
one uses the Cauchy integral theorem. Since matrix (Ktj) is positive definite,
the poles uv are located on the real axes. For t > 0, Г is the semi-infinite
half-circle in the lower half-plane, and for / < 0, Г is the semi-infinite
half-circle in the upper half-plane. With this Г, the initial condition G, = 0 for

60 Chap. 1 General theory of eigenstrains
t < 0 is satisfied. Thus, we have
Gu(x, t) = Ш i- »ВЬа
(9.16)
where H(t) is the Heaviside step function. We introduce an auxiliary function
Ftj{x, t) defined as
Then,
(9.18)
Since the factor of ехр(г'|?- л:) in the integrand of (9.18) is an even function of
?, it can be written as
УАТГ) р „-Г
(9.19)
where ы„ = ?<о„. By using
f°° sin mx cos «x , . , , ч , \ •. ,
/ ax = j7r{sgn(w + n) + sgn(m — n)) form, и > 0,
(9.20)
we have
CO..

* 9. Dynamic Green's functions 61
Since
j v»6" "> -"\")i ("-22)
(9.17) becomes
where
is used.
* Isotropic materials
In this case,
(9-24)
(9.25)
and from (9.12)
<H = ?,V?2> (9-26)

62 Chap. 1 General theory of eigenstrains
and from the second equation in (9.13)
-U=SiJ* ~^J. (9-27)
Substitution of the above result into (9.23) leads to
ds(i)
The following integrals are used:
c2t-x
P Mo
where
c2 =
(9.29
(9.30
When W t - i ¦ x = 0 we have
V P

* 9. Dynamic Green's functions
63
P x1
(9.31)
Therefore, when ^\ —;— t — ? ¦ x = 0,
t/x \ xiXj
Finally, we have
(9.32)
H{t) Э
3?
Я x
(9-зз)
or
(9.34)

64 Chap. 1 General theory of eigenstrains
The above result was first obtained by Stokes A849). The static Green's
function E.8) can be obtained as
ij(x) = fGij(x,t-t')dt'
J — oo
where f = f - t', З/Эг = -9/3r'. The last result agrees with E.8).
* Steady state
If a unit body force in the ^„-direction with time dependence exp(-io:t) is
applied at the origin of coordinates, the /th component of a displacement field
at point x can be expressed by gim(x) exp(-zw?). gim(x) is called the
steady-state elastic wave Green's function. It satisfies the equation of motion
Cljklgkm,,j(x) + P<»2gim(x) + SimS(x) = 0. (9.36)
The solution can be written as
giJ(x) = B7Г) Г СцИ, со) exp(if-x) d|, (9.37)
^ — oo
where G,y(|, w) has been defined by (9.6) or (9.11). For isotropic materials,
(9.26) and (9.27) yield

* 10. Incompatibility 65
with
a2 = pW2/(X + 2M), Р2 = ри2/1л. (9.39)
Then, we have
(9.40)
where * = E,7jc,jc^I/2.
For the two-dimensional case, (9.40) is integrated with respect to хъ from
— oo to oo. Then we have
(9.41)
where H^\z) is the zero order Hankel function of the first kind and
r = (x2 + jc2I//2
* 10. Incompatibility
Eigenstresses are caused by the incompatibility of the eigenstrains. Uniform or
linear distributions of eigenstrains throughout a free body (compatible eigen-
eigenstrains) do not introduce any eigenstress in the material.
The incompatibility of c* has been discussed by Reissner A931) and
Nemenyi A931). Dislocations due to the incompatibility have been studied by
Volterra A907), Weingarten A901), Cesaro A906), and Moriguti A947) from
the viewpoint of elasticity theory in connection with the multiple-valueness of
displacements and rotations. A plasticity theory viewpoint has been developed
by Kondo A955), Bilby A960), and Kroner A958). In this section we follow
the work of Moriguti.
Let us consider an elastic body D which is free from external force but is
subjected to a distribution of eigenstrain e*. Imagine a slender rod AP inside
the elastic body (see Fig. 10.1). If the rod were isolated from its surroundings
and set free, it would change its form slightly and take on a form such as AP',
where one end of the rod, A, is fixed (see Fig. 10.1). If a Cartesian coordinate

66 Chap. 1 General theory of eigenstrains
D
Fig 10 1 Liberation of an imaginary slender rod AP in a continuum D gives deformation AP'
is attached at point P, it will rotate in a different direction. Both displacement
м, and the rotation to, at P are caused by the release of elastic strain etj along
AP.
Let us try to express these components of displacement and rotation as
functions of e,y in the form of line integrals along AP. Since the deformation
of AP to AP' is completely elastic with displacement м,, we can write
te3
x/ 8x,
A0.1)
55,
x, 8xj
where 8м, is the change of м, along a small line element ds = (8x,8x,I/2
along АР (8й, is caused by the release of ds). Since м, is path-dependent, it
does not have the usual derivatives м, , but it has 8н,/8х, which are defined
only along AP. The elastic strain components e,, have been caused by «f,-,
which are piece-wise continuous functions with derivatives. The equations in
A0.1) easily lead to
8Hl=-(eu-uIJ)8xJ. A0.2)
We have also
__1_/_Э_8м^ Э Щ
kJ<l 2 I Эх, 8х, Эх, 8хк ,
1 "[ A0.3)
Э 8ик Э 8ut
Эх, 8х, Эх, 8хА

* 10. Incompatibility 67
and therefore
(ek i~ekt )= ~ ")> (Ю-4)
where
1. а Яп,
k- A0.5)
3 8пк Э
is used. Multiplying A0.4) by Sxk, we have
(ekJ,i-ekiJ)8xk = S<SIJ A0.6)
or
8uij='ehiJ<hlmekm.l8xk-
From A0.7) and A0.2),
with the initial conditions
m,. = wo. = 0 at^. A0.9)
Consider a special case where the end point P coincides with the starting
point A (see Fig. 10.2). Then we have strain release along a closed curve с Let
us call the displacement and the rotation at the end point due to the strain
release a dislocation and a disclination, respectively, as shown in Fig. 10.3. The
end point shifts to A', and the coordinate system x- (fixed to the end point)
rotates to the coordinate system x't after the release. The disclination is
A0.10)

68
Chap 1 General theory of eigenstrains
Fig 10 2 End point P in Fig 10 1 coincides with the starting point A The closed loop and the
surface bounded by the loop are denoted by с and S respectively
and the dislocation is
M«= -
A0.11)
The closed curve с is called the Burgers circuit. The integrations are defined
Fig 10 3 Dislocation and disclination are relative orientations between the x- and x'-coordinate
systems

* 10. Incompatibility 69
on the closed curve с before the release. Applying Stokes' theorem to A0.10),
we have
п„ = jj4ijRhqnq dS, A0.12)
where nq is the unit normal on dS, and
A0.13)
The symmetric tensor Rhq is called the incompatibility tensor.
Before we apply Stokes' theorem, expression A0.11) is slightly modified.
This is necessary since п^ in the integrand A0.11) has no derivatives.
Integration by parts of A0.11) and the definition of A0.7) lead to
+ uijX° - фх^ниеН1тект1 dxk, A0.14)
c j c
where xj is the coordinate of point P( = A) in the ^-coordinate system before
the release. The initial conditions A0.9) have been considered. The last
expression of A0.14) is called Cesaro's integral A906). By the use of Stokes'
theorem, A0.14) can be simplified as
4ijRh4»4 ds- A0-15)
Since the components of the vector Od' are the sum of ut and the effect of the
rotation ujj at A', they can be expressed as
i, = щ- пиху = й,. + tIJkujxl A0.16)
Then,
The result A0.16) corresponds to G.19). [«,], f.ipqu>pxq, and -eipqwpx° in
G.19) are equivalent to ^, f.ijkUjX°k, and u, in A0.16), respectively, and с is the
Burgers circuit.

70
Chap 1 General theory of eigenstrains
Fig 10 4 In a multiply connected material, the Burgers circuit с may contain a cavity
An elastic body subjected to an applied load has a compatible elastic strain
and er= \{ut + Uj,). In this case, Rhq becomes zero when it is substituted
into A0.13). Compatible elastic deformations do not cause any dislocation or
disclination. If an elastic body has an internal stress (eigenstress) field due to
an eigenstrain c* the dislocation and disclination defined by A0.17) and
A0.12) are not zero. The incompatibility caused by ?* can be calculated by
substitution of
into A0.13). Then, we have
A0.18)
A0.19)
Equations A0.12) and A0.17) are expressions for disclination and dislocation
as functions of the eigenstrains.
The surface S in A0.12) and A0.17) should be bounded by the closed curve
с and should consist of material points. When a multiply connected material is
considered and a closed curve contains a cavity (see Fig. 10.4), the line
integrals A0.10) and A0.14) defined on с cannot be transformed into A0.12)
Fig 10 5 Dislocation

* 10. Incompatibility 71
(а)
Fig 10 6 Disclination
and A0.17). Therefore, dislocations and disclinations can exist even if Rhq = 0.
It is easily shown that any two closed curves surrounding the same cavity
define the same dislocations and disclinations if Rhq = 0. The proof can be
shown by applying Stokes' theorem to the domain bounded by the two curves.
Examples are Volterra's dislocation (Fig. 10.5(b)) and disclination (Fig. 10.6(b))
which are made by welding Figs. 10.5(a) and 10.6(a), respectively.
* Riemann-Christoffel curvature tensor
When the Burgers circuit is taken as an infinitesimal small loop, formula
A0.17) can be written as
i^-XjtujR^US. A0.20)
Furthermore, we put
Then, A0.20) can be written as
where
**,„;,• = «*,¦/,«»**,• (Ю.23)
Rmnji is called the Riemann-Christoffel curvature tensor and Rh the Ricci
tensor. Although the eigenstrain e* in A0.18) is incompatible in the three-di-
three-dimensional Euclidean space when Rhq Ф 0, it might be compatible in the
three-dimensional Riemannian space. The curvature of the Riemannian space

72 Chap. 1 General theory of eigenstrains
(o) <t»
Fig 10 7 Two-dimensional Euclidean space (a) and two-dimensional Riemanman space (b)
Plastic deformation can be characterized by the transformation from (a) to (b)
is expressed by Rmn]i. «*
the Riemannian space with the fundamental metric tensor gtJ, that is,
<fy = *(&,-«„)¦ A0-24)
When A0.19) is substituted into A0.23) by use of A0.24), we have
mnji llomi.nj onj,im &mj,ni oni,mj) v /
which is nothing more than the definition of the Riemann-Christoffel curva-
curvature tensor for small strains (see Sokolnikoff 1964, pp. 89 and 91).
In order to explain the physical meaning of the above result, let us consider
a two-dimensional case for simplicity. Figure 10.7(a) shows a two-dimensional
Euclidean space subjected to an eigenstress field due to a constraint eigen-
strain e*.. This is the usual dislocated state of an imperfect crystal. Figure
10.7(b) is a two-dimensional Riemannian space with the fundamental metric
tensor 2e*j + StJ. This Riemannian space is an imaginary state of the material
and is stress-free; it is called the natural state. The associated elastic stress and
strain are necessary to make the natural state shown in Fig. 10.7(b) conform to
the physical state shown in Fig. 10.7(a). The eigenstrain is compatible in Fig.
10.7(b), but it is constrained and incompatible in Fig. 10.7(a). The sum of the
elastic strain and the eigenstrain is, of course, compatible in Fig. 10.7(a).
It is important to note that the two-dimensional Riemannian space is
embedded in the three-dimensional Euclidean space. Similarly, a three-dimen-
three-dimensional Riemannian space can be embedded in a six-dimensional Euclidean
space. If a deformation is defined from the strained state (Fig. 10.7(a) for the
two-dimensional case) into the natural state (Fig. 10.7(b) for the two-dimen-
two-dimensional case), the deformation can be characterized by a deviation from one

* 10. Incompatibility 73
space into a higher dimensional space, as seen in the buckling phenomena of
elastic plates. Kondo A949, 1955) used this buckling analogy in his explana-
explanation of the yielding phenomena of mild steel. He postulated that yielding is a
buckling phenomenon of a three-dimensional Euclidean space into a six-di-
six-dimensional Euclidean space (three-dimensional Riemannian space). The Rie-
mann-Christoffel curvature tensor or the Euler-Schouten curvature tensor is
the geometric object in Kondo's theory of plasticity. On the other hand, Bilby,
Bullough, and Smith A955), Bilby A960), Kroner A956, 1958), and Kroner
and Rieder A956) adopted the distant parallelism in a linear connection and
described the Burgers vector by Schouten's torsion tensor. A more general
expression for the Burgers vector has been given by Kondo A955).

74 Chap. 2 Isotropic inclusions
Isotropic inclusions
Explicit formulae are derived for elastic fields caused by inclusions. Most of
the cases considered in this chapter are ellipsoidal inclusions in an isotropic
infinite body, and the elastic moduli are the same for inclusions and matrices.
The case when inclusions and matrices have different elastic moduli is treated
in Chapter 4.
Eigenstrains in the inclusions are given by constants or by polynomials of
coordinates. A special emphasis is placed on the case of constant eigenstrains.
11. Eshelby's solution
An ellipsoidal inclusion ?2 is considered in an isotropic infinite body. Eigen-
Eigenstrains given in the ellipsoidal domain are assumed to be uniform (constant).
The solution of the problem has been investigated by Goodier A937) in the
case when eigenstrains are thermal expansion strains. For general eigenstrains,
the solution has been given by Eshelby A957, 1959, 1961). Expressions for the
solution are different for interior points (points inside the inclusion) and
exterior points (points outside the inclusion). Eshelby's most valuable result is
that the strain and stress fields become uniform for the interior points.
From F.1) we have
iJ<k(x-x')dx', A1.1)
where Q (see Fig. 11.1) is given by
x\/a\ + x\/a\ + x\/a\ < 1 A1.2)

//. Eshelby's solution
75
Fig 11 1 An ellipsoidal inclusion with principal half axes «b a2, and a,
and G,j(x - x') is, from E.8),
x-x
'I 2
After some manipulation we obtain
,., dx'
- x\
where
3/,./,/,.
A1.3)
A1.4)
A1.5)
The vector / is a unit vector (x' - x)/| x' - x |.
Interior points
When point x is located inside the inclusion, the integral in A1.4) is explicitly
performed. As shown in Fig. 11.2, the volume element dx' in A1.4) can be
written as
dx' = dr dS = drr2 dw, A1-6)
where r = | x' - x | and dw is a surface element of a unit sphere ? centered at
point x. Upon integration with respect to r, we have
A1.7)

76
Chap. 2 Isotropic inclusions
Fig 112 Q is an ellipsoidal inclusion, 2 is a unit sphere centered at point x
where r(l) is the positive root of
(*, + rlxf/a\ + (x2 + rl2f/al + (x3 + rl3f/a23 = 1,
that is,
where
A1.8)
A1.9)
f= lxxja\ + l2x2/a22 + l3x3/aj,
e=l- x\/a\ - x\/a\ - x\/a\.
A1.10)
When A1.9) is inserted in A1.7), the term (f2/g2 + e/gI/2 can be omitted,
since it is even in /, while gijk is odd. To retain the advantages of suffix
notation, we introduce the vector
= l2/a2,
Then,
and the strain components become
f * /* Л J? - ~г Л Я-
l-v)Jy g
A1.11)
A1.12)
A1.13)

11. Eshelby's solution 11
The integral in A1.13) is independent of лг. Therefore, we have an attractive
conclusion in that the strain (and therefore the stress) is uniform inside the
inclusion. The surface integrals of the type f^l^l^g'1 dw can be reduced to
simple integrals according to the work by Routh A895),
ll&w /•«> ds
a{g Jo {af + s) A(s)
If dw /•«> ds
—^-=2чта1агаг\ — , A1.14)
afg Jo (ai + s) A(s)
l\l\ dw /-00 ds
JTT- = 1VW3] , 2^ w 2л_ ч ., ..
a{a\g Jo (af + s)(a2 + s)A(s)
with A(i) = (flfi+ iI/2(«2 + *I/2(аз + s)l/2. The remaining coefficients are
found by the simultaneous cyclic permutation of A, 2, 3), (alt a2, a3) and
(A, 12> h)-
It is convenient to write A1.13) as
^j = SiJkfi*kl. A1.15)
Then,
Snu =
^
1
1
8тг
1
8тг
1
-2v
-2v
-2v
-2v
Sl133 =
-v) In
All other non-zero components are obtained by the cyclic permutation of
A, 2, 3). The components which cannot be obtained by the cyclic permutation
are zero; for instance, S1112 = ^1223 = ^1232 = 0- Sjjki ls called Eshelby's tensor.

78 Chap 2 Isotropic inclusions
The integrals in A1.14) are expressed by the standard elliptic integrals
(Gradshteyn and Ryzhik, 1965). Assuming ax > a2 > a3,
( 2_ 2 А1/2
т —1*-з , Дг^а! а3) ,. ч.
where
' Е{в>к) = Л1 -kl sin2wI/2 dHI'
¦'o (l-^2 sin2wI
A1.18)
e = sm-i(l-ai/a?)l/2, к = {(a* - aj)/(a? - aj)}1/2.
The definitions for g and / and A1.14) yield the following formulae:
The last formula in A1.19) is obtained when we split the factor (a2 + s)~l(a\
+ s)-1 in the integral A1.14) for /12 into partial fractions. Equations in
A1.19) and their cyclic counterparts give sufficient relations to express /, and
Ijj in terms of Ix and I3.
Finally, the stress components become
+

//. Eshelby's solution 79
\-2v
A1.20)
and other components are obtained by the cyclic permutation of A, 2, 3).
The integrals A1.17) become elementary functions for special shapes of
inclusions, as listed below.
Sphere (a, = a2 = a3 = a)
¦Sim
^1212
From
'22 = /33
2222
= ^2233 =
•$2323
A1.20)
1,. '
= hi = h
3333 ^
- s33U = s
3131 j
we have
16
1 Л 11
1 — v )
r*
7-
5A
1133
4-
¦5A
i
Ai =
-5»»
-5*>
5j
'l5(
= 4t7/5 a2,
!211 — ^3322 =
I-L2
(n21)
_ \ '
A1.21.1)
15A - у) 12
All other stress components are obtained by the cyclic permutation of A, 2, 3).

80 Chap 2 Isotropic inclusions
Elliptic cylinder (a3 -* oo)
1х=Атта1/(а1 + а2), I2-= 4ira1/(a1+ a2), /3 = 0,
/12 = 4w/( «i + «2 J> 3/n = 4w/fl? - Аг >
1-/12> /13 = /23 = /33 = 0,
2(l-iO\(ei + e
2A -
9 -0 9 1 / al + a2 1-2,
3322" 5+
cn
5 = 'h ; § = 1 ;
and
?33'

11. Eshelby's solution 81
2 +
¦
a2
el2> a23= /x«1+a2
Penny shape (a, = a, s> a3)
11=12 = 'п2аъ/а1, I3 = 4w — 2m2aJ/a1,
Аз = 723 = 73i = 7з2 = 3(|тг - 772а3/а1)/а12,
hi = 722 = Зт72а3/4«3, /33 = f 77/а2;
32A -,)%,' 6з333 ~ 1 1-р 4 а/
1122 ~ 22И ~ 32A - i.) вх ' 1131 ~ 2233 ~ 8A -у)
I 4v +1 a3
1212 ~ 32A - i.) ax ' ^1313 ~ -^2323 -
e c 1-2» 77 a3 v 1 -2f

82
and
/Т /1 II
when a 3 -
/ .si
2p •+•
Q / I
ol J. —
2p +
8A-
oi X —
/1 / T -у \
l '
1
7Г
С
1
L6(
a €*3>
WO3 2 г
«г ?n 8A
h «23. CT3i/2/
I-»7) al
13
16^-
+ 1 ¦
-)
1 4(
7T(
1 mt
v) a
v-2
1 - v
Chap.
2з e* + li
¦1 tn + 32
2 4A-1
3 *
) a\
2 Isotropic 1
(.1-»') ai
13 wa3
I
1таъ
v) a\
A1.23.1)
/!=/2=0, /3=477,
4т7-/3
A1.23.2)
_ I
~ 2
^3333 = 1 > and all other 5,- -л/ = О,
and
A1.23.3)
a33 = 0, a31 = 0, a32 = 0.

11. Eshelby's solution 83
Flat ellipsoid (a,> a2 » a3)
I1=4na2a3{F(k)-E(k)}/(a2-a2),
12 = 4va3E(k)/a2 - 4-na2a3{F{k) - E{k)}/(a2 - a\),
13 = 4ir-4ira3E(k)/a2, A1-24)
I12 = [ЛяаэЕ(к)/а2 - iva2a3{F(k) - E(k)}/{a^ - a22)]/(a^ - a22),
I23 = [4w- i«a3E(k)/a2 + 4*a2a3{F(k) - E(k)}/(a2 - a\)\/a\,
hi = I4" ~ 4™2a3{F(k) - E(k)}/(a2 - a\) - 4<na3E(k)/a2\/a2,
I33 = 4v/3a2,
where F(k) and E(k) are the complete elliptic integrals of the first and the
second kind, respectively,
o
f/2(
o
F(k) = f/2(l - k2 sin^)/2 <1ф, (H-25)
•'o
k2 = {a2-a2)/a2.
For €* = «*3, A1.20) becomes
o33/2ti= -a3e*33E(k)/2(\ - p)a2. A1.26)
For?*=?*3,
\()] A1.27)

84 Chap 2 Isotropic inclusions
Oblate spheroid (at = a, > a})
1 2
B2)
2 / / 2 \ 1-/2\
4(а32-д2)'
Аз = Аз = (А - А)/(«з2 - «?), 3/33 = 4V«32 - 2/,з.
Prolate spheroid (a, > а2 = а})
1/2 \
i.uu,a% а, at
I = г = L^ _L _1 _ 1
3/22 = 4^/«22 - /23 - ('2 - A )/( «I2 - «22 ).
A1.28)
Exterior points
When point д: is located outside the inclusion 12, a slightly different approach
is employed. When expression E.9) for GtJ is substituted into A1.1), we have
{% 2*,,/- 4(! - ")««,/}' (И-30)

11. Eshelby's solution 85
where
%(x)=e*Jjx-x'\dx', Фи(х)=€Г^1/\х-х'\йх'. A1.31)
From A1.30) the induced strain field can be written as
,ij - 2A - »)(*,*» + Фук,к1)} A1.32)
or
^tM) = DiJkl{x)tt,, A1.33)
where
8^A - v)Dijkl(x) = *,„,., - 2v8kl4>tiJ
'A - ")[Ф,*>8,7 + Ф.*/*,/ + Ф./А* + Ф,/,«д], (H-34)
and
A1.35)
As will be shown in the next section, Ferrers A877) and Dyson A891)
expressed the above integrals in terms of the following elliptic integrals:
(af
d' A1.36)
) 4()

86 Chap. 2 Isotropic inclusions
where A(s) = {(a? + s)(a\ + s)(aj + s)}1/2, and Л is the largest positive root
of the equation
x\/{al + X) + x\/{a\ + X) + x\/{a\ + X) = 1 A1.37)
for an exterior point x. X = 0 for interior points of ?2. The result is
where the following summation convention has been used: repeated lower case
indices are summed up from 1 to 3; upper case indices take on the same
numbers as the corresponding lower case ones but are not summed. For
example,
xkIK = xlIl where k = \,
= x2l2 where A: = 2, A1.39)
S./y = /2 if i =./ = 2,
= 0 if i Ф).
Equation A1.34) requires higher derivatives of <j> and \p. Since the lower
bound of the integral limits in A1.36) is only a function of x, the derivatives of
/, /,., I-j can be reduced to the derivative of X. From A1.37) we have
X,--^-/—5Л-. A1.40)
The following obvious relations are useful for further calculations: Consider
the derivative

11. Eshelby's solution 87
from which it follows that
Using A1.39), it can be shown that
7
,,irr ^ >, (n-40-3)
It can be easily seen from A1.40.3) that
-xrxrIRij k(X)} =-2хч1ви k(\). A1.40.4)
The above result A1.40.4) is used for derivatives of A1.38). Then, we have
A1.40.5)
-(SuxJ + 8J,xi)[lJ(\)-allrj(\j\,k-xlxJ[lJ(\)-allfJ(\j\tkl.
A1.40.6)
Finally, we have, from A1.34),
- r)SiJkl(\) + 2v8klXiI7J(X)
- v){8ilXkIKJ(\) + 8jlXkJK!l(X) + 8ikx,ILJ(X) + 8jkx,IL,{X)}
A1.41)

88 Chap. 2 Isotropic inclusions
where
iit(\-v)Sijkl{\)
= &iJ&kl[2vII(\)-IKi\)+a2IIKI{\)\
+ {8ik8jl + 8jk8,l){a)lIJ{\)-IJ(\) + A -»0[/*(A) + /JX)]}. (H-42)
The above result holds for both exterior and interior points. For interior
points, X = 0 and all derivatives of /, and /; • vanish. Then Dijkl becomes
Sjjkl@), which can be found to be equal to A1.16), where the relation A1.19)
is used to establish this equivalency.
Recently, Tanaka and Mura A982) have proposed an alternative method to
evaluate the elastic field for exterior points. First, obtain the stress field for the
interior points of ?2 and denote it by а{] (in). Next, find the stress field for the
exterior points, assuming that ?2 is a void and the applied stress is — a,-y(in).
Denote the stress field by a/7(out). Then, the stress field for the exterior points
in the original problem is the sum of a, (out) and a,, -(in).
Thermal expansion with central symmetry
The thermal stress problem in an isotropic infinite body under a temperature
T symmetrical with respect to a point, has been well-known and its solution is
found to be
A1.43)
(e.g., Timoshenko and Goodier 1951, p. 417) where r is the distance from the
center of symmetry, и is the displacement in the r-direction, and a is the
thermal expansion coefficient.
When T is constant inside a spherical domain ?2 with radius a, and zero
outside the sphere, we have
for interior points of ?2, and

* 12. Ellipsoidal inclusions with polynomial eigenstrains 89
2 1 + y a3 1 l + v a3 _ (лл ...
Г Т A145)
for exterior points, where cr and it are the radial and tangential strains.
It will be shown that the same result can be obtained from A1.33) with
A1.41), where e* = S^aT.
The following calculation is straightforward. From A1.37), X = r2 - a2.
From A2.19) in the next section, IX(X) = I2(X) = /3(X) = 4na3/3r3, /,7(X) =
4<паъ/5rs. For interior points, X = 0 and I1= I2 = I3 = 4w/3, IfJ = 4ir/5a2.
When these results are substituted into A1.41) and A1.42), we have A1.44) or
A1.45), depending on the position of дг.
* 12. Ellipsoidal inclusions with polynomial eigenstrains
An ellipsoidal inclusion is considered in an isotropic infinite body. Eigen-
Eigenstrains in the ellipsoidal domain are given in the form of polynomials of
coordinates such as
<?>(*) = BiJkxk + BiJklxkx, + ¦¦¦, A2.1)
where Bijk, Bijkl,... are constants symmetnc with respect to the free indices /,
j and Bijkl = Bijlk, Bijklm = Bijkml, etc. The constant term is excluded in A2.1)
since it has been discussed already in the last section.
In this section, we follow to a large extent the Ph.D. theses of Sendeckyj
A967) and Moschovidis A975).
Both displacement fields for interior and exterior points of ii can be
expressed by A1.30),
where
*;,.(*)= jH *-x'| ?*(x')dx' A2.3)
and

90 Chap. 2 Isotropic inclusions
Ф,у(дг) and ^j(x) are the harmonic and biharmonic potentials due to a body
$2 of density «,*•(*'), and have the following properties:
and
0 for jc out of J2.
The strain field can be expressed by A1.32). When A2.1) is substituted into
A2.3) and A2.4), we have
% = Викфк + Вок/фк/+ ¦¦¦ A2.7)
and
% = ВикФк + Вик1фк1+..-, A2.8)
where the following definitions have been made:
',x'j- x'kdx>
^ and ф have the same definitions as A1.35). Substitution of A2.7) and A2.8)
into A1.32) leads to
«,->(*) = Dukiq(*)Bkiq + Dljklqr{x)Bklqr + ¦ ¦ ¦, A2.10)
where
8WA - v)Dljklq = ^qklij - 2v8kl4>4tlj
- A - ")[Ф,.*А + Ф,.*.Л/ + Ф,./>8.-* + Ф,,«8,*]' A2.11)
etc.

* 12. Ellipsoidal inclusions with polynomial eigenstrains 91
The expression for Dijklqr is obtained from the above by replacing \pq and
% ЬУ ^qr and <V> respectively.
As seen in the next subsection, the harmonic potentials in A2.9) can be
written in terms of the following elliptic integrals:
(m2)
/ U(
Vn(x) = тга.а.а, f У
(a2 + s){a] + s)A(s) '
etc.,
where A(s) = {(a2 + s)(a\ + s)(a2 + s)}l/2, U(s) = 1 - {x2/(a^ + s) +
x\/(a\ + s) + xj/(aj + s)}, and \ is the largest positive root of A1.37) for an
exterior point л:. Л = 0 for interior points of Q. According to Dyson A891),
A2.9) can be written as
<S>n = a2NxnVN, A2.13)
Kn = a2M{xmxna2NVMN+ \8mn[v - xrxrVR- a2M{VM- xrxrVRM)}} ,
and
*„.,¦= -*«/Х{(Р'- хг*УК) - a%(VN- xrxrVRN)}
+ a2NxnXi(V,-a2NVIN),
A2.14)
tmnJ = a2Ma2N { -\{8mixn + 8mxm)[vM- xrxrVRM - a2N(VNM - xrxrVRMN)]
+ и2„8тпх,{(Г- xrxrVR) - (a2 + a2M)(VM- xrxrVRM)
+ a4/(VM!-xrxrVRMr)}.
In the above expressions, the summation convention demonstrated by A1.39)

92 Chap. 2 Isotropic inclusions
was used. By definition, the F-integrals in A2.12) and the /-integrals in A1.36)
are related,
V(x) = \{I{\)-xrxrIR{K)},
Vi{x) = \{Ii(\)-xrxrIRl{\)), A2.15)
Vlj{x) = \{lij{\)-xrxrIRlJ{\)},
etc.
When jc is an interior point of i2, then \ = 0, and /, /,, /,-_,-,... are
constants and all their derivatives vanish. Thus, the first derivatives of the
F-integrals are linear functions of x, and the second derivatives are constants.
Therefore, Dijklq{x) becomes a homogeneous linear function of л: and
DiJkl4r{x) becomes a sum of constants and quadratic functions of x, that is,
for interior points x,
DlJkl4{x) = DiJklqtm(o)xm,
DiJklqr(x)=DtJklqr@) + \Dljkl4rtmn@)xmxn.
The x dependency of higher order DiJkl... can be similarly investigated, and it
is concluded that «,-,-(x) for interior points becomes an inhomogeneous
polynomial of x whose terms are of degree n, (n — 2), (и — 4),..., if ?*/(*) is
a homogeneous polynomial of дс with degree n. This general conclusion was
first pointed out by Eshelby A961). Asaro and Barnett A975) reached the
same conclusion for anisotropic materials.
* The I-integrals
The /-integrals in A1.36) can be expressed by the first and second elliptic
integrals defined by A1.18); for a1 > a2 > a3,
1/2, к),
aiyl/2{F{e,k)-E@,k)},
A2.17)
\1

* 12. Ellipsoidal inclusions with polynomial eigenstrains 93
1a2a3(al-aj) \a2-aj)
where F, E, в and к have been defined in A1.18).
The other higher order /-integrals can be expressed by /,, using the
following relations:
/2(X) + /3(X) =
3/n(X) = 4^1a2a3/(«12 + X) A(\) -
A2.18)
5/m(X) = А-паха2аг/{а1 + XJ A(X) -
where
The cyclic permutation with respect to A, 2, 3) can be applied in A2.18).
The /-integrals become elementary functions for special cases as shown in
the following.
* Sphere (a, = a2 = a} = a)

94
Chap. 2 Isotropic inclusions
* Elliptic cylinder (a3—> со)
-i ,
1
/3(X) = 0.
* Oblate spheroid (a,= a2> a3)
*l - A
IT
— — arc tan
al-a\
arc cos
= /2(X) = 1<па\аъ{пс cos ft - bd)/(a\ - a\)V~'\
A2.20)
| - arc cos
flJK/a,
where ft = у (a2 + X)/(a2 + X) and rf= у (a2 — a3)/(a2 + X).
* Prolate spheroid (a, > a2 = a3)
t—i г
Аттаха\
/(X)=-
In
- /a
? - a
A2.21)
i/fl? - a\
- arc cosh ft,
K + X-
/X(X) = 4тга1а2(агс cosh ft - d/b)/(a2 - ajf/2,
i = lira^Kbd- arc cosh ft)/(a2 - a3) ,
A2.22)
where ft = /(a2 + X)/(a32 + X) and d= 1/@? - aj)/(a

* 12. Ellipsoidal inclusions with polynomial eigenstrains 95
Table 121 Limits of the /-integrals as a3 -»0 for X Ф 0 and X = 0.
Iim( ¦•)
<J3-»O
Ijj (X) with i, j,. #3
3 v /
13 v /
23^ /
Лз(Л)
X#0
0
0
0000
x = o
0
4w
4j/uJ
4тг/а2
00
4w/3
/n3(X) 0
/223(X) 0
/,^(X) 0
2
2
0 00
0 00
0
0
0 00
О 4тг/5
0
0 4V
0 4m/a\a\
0
* Elliptical plate (aj>a7>a}, a}^>0)
lim / (X)=0 for /, у,... ФЪ. A2.23)
Other cases are listed in Table 12.1.
* The Ferrers and Dyson formula
Ferrers A877) and Dyson A891) investigated integration of the harmonic
potential of an ellipsoid,
Their results have already been used in A1.38), A2.13), and A2.14).

96 Chap 2 Isotropic inclusions
A general result, given in Dyson's paper, states that if
p(x)= m (l-^r Vf —,— >—) form>0, A2.25)
чта1а2аъ \ a) J \ ax a2 a3)
then
U+ a^ + a2 + /4(i)
where
Of + S
A2.27)
aj + s Э Э
a 2 Эх,. Эх,-'
Furthermore, Л is the largest positive root of
*'*' =1; i.e., ?/(Л) = 0. A2.28)
The summation convention in A1.39) has been used in the above equations.
All integrals in A2.9) can be expressed in this type of integral:
, х]х) ¦ ¦ ¦ dx'
since | jc — jc'| = (xixi - 2x,x,' + x\x\)/\x- x'\. Comparing A2.25) with xtXj
• • •, we have m = 1 and
V s"Un(s)
и?о22,и!(л+1)!
^. A2.30)
(s)

13. Energies of inclusions 97
When the differential operator L" in A2.30) is performed, the required
integrals shown in A1.38), A2.13) and A2.14) can easily be obtained.
13. Energies of inclusions
Consider a finite body D containing inclusions Q. The material is homoge-
homogeneous, and anisotropic or isotropic. ?2 is the sum of domains occupied by the
inclusions. The volume of ?2 is denoted by V.
Elastic strain energy
First we consider the case when the body D is free from any external force and
surface constraint, but eigenstrains e,* are prescribed in ?2. The (elastic) strain
energy is
W* = \(o e dD, A3.1)
JD
where etJ = etJ - e* and e/y = {{ut 7 + Uj,). Integrating by parts we obtain
joijUi4j dD = jaIJu,nJ dS - JiaijJui dD = 0, A3.2)
since oi]n] = 0 (free surface condition) on S and oihJ = 0 (equilibrium condi-
condition) in D, where S is the boundary of D and и, is the exterior unit vector
normal to S. Therefore, we have
A3.3)
If ?2 is an ellipsoidal inclusion and «*, is uniform, a/y in ?2 is also uniform.
Then, A3.3) is written as
W=-\Voi}t*r A3.4)
o-- has been obtained in Section 11 for isotropic, and in Section 17 for
anisotropic materials.
If ?2 is the sum of two inclusions ?2г and ?22, A3.3) is wntten as
U/* — 1 I „A) J. „B) 1 ,0) Jn i / _A) i _B) \ ,B) J r-v (л i л -¦ Ч
^ ~ "г / \aij' + au)eh dD+ /_ К +< M€-V dD ' A3.4.1)

98 Chap. 2 Isotropic inclusions
where c^ and if) are eigenstrains given in Qx and J22, respectively, a^ is the
stress caused by eff, and a/y2) is the stress caused by tf). It is shown that
A3.4.2)
as follows. Since eft = 0 in D - U, it holds that
P)(?<)>-«W + <))dD, A3.4.3)
where иР} is the displacement caused by «5]}. We know that afj\(^y- u?)) =
-offefy and jrpj-f'uty dD = 0, where ejy is the elastic strain corresponding
to ст/7г). Therefore, we have
UD=-lo<f>e<»UD. A3.4.4)
Similarly,
fVef?*D. A3.4.5)
Since а/72)е^' = Cijkle$e}y = ojtfeffi, the equality A3.4.2) is proven. Due to
this fact, A3.4.1) is written as
w* = - 2 \ a/-MV dZ> + / 0Ш1) dD + 2 оШ? dD . A3.4.6)
Consider now the case when the body D, containing inclusions ?2, is
additionally subjected to external surface tractions Fr The displacement field
is the sum of м,° and и,-, where uf is the displacement if Ft act alone in the
absence of eigenstrains, and i/, is due to the eigenstrains prescribed in the
inclusions. Then, the (elastic) strain energy is
A3.5)

13. Energies of inclusions 99
where a° = CiJkluokj. Since aiJJ = 0 in D and a.y«7 = 0 on S, the integration
by parts gives
0. A3.6)
It is also seen that
0, A3.7)
since a°(Ulj - €* ) = Cljkluokl(UiJ - e* ) = и° ,oH = а,ум°у and /Da,X/
0.
The elastic strain energy, therefore, becomes
= hjfiulj UD - \J4tfj dD. A3.8)
W*
It is interesting to note that the elastic strain energy is the sum of the two
energies caused respectively by Ft and e*y. This is a Colonnetti's theorem
which will be discussed again in Section 25.
Interaction energy
The total potential energy of a body subjected to a surface traction F: and
eigenstrains ?,* in S2 is defined by
IV= W*- (FJu° + u,)dS, A3.9)
Js
where а°иу = F, on S, and W* is defined by A3.5). If c* = 0 in D, W
becomes
% = \jofrlj UD - JFtf As. A3.10)
If Ft = 0 on S, Ж becomes
/,< d^>- A3.11)

100 Chap 2 Isotropic inclusions
The interaction energy between e*y and Ft is defined by
AW=W-WQ-Wl. A3.12)
Then, we have
AW = - (FiU, dS= - (o°tf dD, A3.13)
since JsFiUl dS = /8а°л,«, dS = /Da>,,, dD = /Da°(«,,y - e* ) dZ> + /Da,%*
dD = /Da°€* dD due to A3.7).
If Q is an ellipsoidal inclusion and c* is uniform,
AW=-Vo»i*r A3.14)
Under a constant temperature condition, the elastic strain energy of a body is
the Helmholtz free energy of the body. The total potential energy A3.9) is the
sum of the elastic strain energy of the body and the potential energy of an
external force and corresponds to the Gibbs free energy of the body. This
thermodynamic discussion was given by Eshelby A956). One should mention
that A3.11) can also be considered to be the Gibbs free energy of the body
when an external force is absent and an inclusion is a source of the stress field.
A3.10) is the Gibbs free energy of the body when only an external force is a
source of the stress field. Therefore, the interaction term A3.12) is an extra
part of the Gibbs free energy of the body, produced by the coexistence of the
two sources of the stress field. It is often called the interaction (Gibbs-free)
energy between ef and Fr
Suppose that a body D is subjected to Ft alone. If eigenstrains due to
inclusions are introduced in Q, the free energy will increase by
AW= W- Wo. A3.15)
This AW is an interesting quantity, particularly in fracture mechanics, as
explained in Chapter 5. Equation A3.15) can be written as
dD. A3.16)
It should be emphasized that energies A3.3), A3.8), A3.11), A3.13) and
A3.16) are expressed by the quantities defined inside ?2. Energy calculations,
therefore, become easier.

13. Energies of inclusions
101
Elastic strain energies in an isotropic infinite body, which are due to
constant eigenstrains prescribed in inclusions of indicated geometries, are
presented below.
Strain energy due to a spherical inclusion
W* =
where a is the radius of the sphere.
Elliptic cylinder (a3 —> ooj
*з21)}, A3.17)
2A-,)
(al+a2J
с*2
- 11
33
+
Ava,
22 + 4A -
which is defined per unit length of the cylinder.
Penny-shaped flat ellipsoid (a; = a, = a, a3 <sc a)
4A - «0
A3.18)
W* = ^а2
e*22
)/(l -
A3.19)

102
Chap. 2 Isotropic inclusions
Fig 13 1 Strain energy coefficients for an oblate spheroidal inclusion
Spheroid
For a1 = a2 = a, fi = аг/а,
w* = | {(
A3.20)
For аъ = ax = a, /1 = а2/а'
<*22i
A3.20.1)
Numencal values of fe, have been calculated by Shibata and Ono A975) as
functions of /3 for oblate spheroids as shown in Fig. 13.1.

13. Energies of inclusions 103
It is interesting to note that for dilatational eigenstrains e* = 5,/*, the
elastic strain energy per unit volume of inclusions is a constant independent of
the shape of inclusions,
W*/V=2n(e*J(l + v)/(l-v), A3.21)
where V= %ттахагаг.
The above result A3.21) holds for any shape of inclusion Q with volume V.
This is because when e*y = S,ye*, B.7) and A1.33) lead to
a,,= {Bti+3\)DiijJ-Fli+9\)}t*. A3.22)
From A1.34) we have
DUjJ = {l + v)/{\-v), A3.23)
where
*'"' 2Ф; . п A3.24)
ф .. = -Ait in 12,
= 0 outside 12,
are used. Equation A3.24) follows directly from A2.5) and A2.6). Thus, it can
be said that the hydrostatic pressure \ou is uniform for any shape of inclusion
when the eigenstrain is dilatational. It becomes
1 + v
4*ii2
= 0 outside 12. A3.25)
The elastic strain energy A3.3) becomes
W*=-\ot.t*V A3.26)
which leads to A3.21).
The above results A3.25) hold also when c* = 8. e* is a function of x. From
B.8) and A2.2) we have
1-2, (c»
) "

104 Chap. 2 Isotropic inclusions
where A2.5) and Ф, = j5, Фтт are used. Furthermore A2.6) leads to
a.,=
= 0 outside Q. A3.28)
Then, the elastic strain energy A3.3) becomes
w* - T-V /o{'*(*)}2 dZ)- A3-29)
In view of A3.28), two separate inclusions, 52X and fi2, with pure dilatational
eigenstrains, e* = S/y-«*, do not interact. For anisotropic materials, A3.28) does
not hold generally and, therefore, there exists an interaction between two
dilatational inclusions. This problem is discussed by considering specific
examples in Section 26; see also Sines and Kikuchi A958).
If the inclusion is an inhomogeneous inclusion with dilatational eigenstrain
tfj = 8и(.р, the elastic strain energy depends on the shape of the inclusion even
if the elastic moduli of the matrix and inhomogeneity are isotropic. As seen in
B2.29) or B2.30), the equivalent eigenstrain «** defined in B2.14) and B2.15)
is not purely dilatational. Because of this, the hydrostatic stress in the
inhomogeneous inclusion depends on its shape and A3.26) is not valid. These
facts should be kept in mind when one discusses the elastic state of precipi-
precipitates.
* 14. Cuboidal inclusions
The elastic fields due to a cuboidal inclusion with uniform eigenstrains have
been calculated by Faivre A964), Sankaran and Laird A976), Lee and Johnson
A977), and Chiu A977). The solid is assumed to be an infinite isotropic elastic
medium. The calculation is interesting, because it shows how the stress
distribution for interior points deviates from the result for the ellipsoidal
inclusion where the stress distribution is uniform.
The cuboidal inclusion is shown in Fig. 14.1. Expression F.2) is used, and
t*j is constant in the cuboidal domain ?2. The integral in F.2) with respect to
*¦' is easily performed and we have
8
u (x) = (i/87r3)c* У (-1
p~\
exp(j'?-c ) d?, A4.1)

* 14. Cuboidal inclusions
105
— - 2o, —>
Fig 141 Cuboidal inclusion
where
c3 =
x2-a2, x3-a3),
x2-a2, x3-a3),
x2 + a2, x3 - a3),
- a3),
x2 + a2,
x2 + a2,
x2-a2,
x2-a2
A4.2)
+ «э).
+ а3),
с8 = (хг + аг, х2 + а2, х3 + а3).
The following integrals are used for the integrations A4.1), and a detailed
calculation is given by Chiu A977):
<n2xxx2
¦ x)
<n2xlx2x3
2*2 + *?
— w ;
R3
A4.3)

106
Chap 2 Isotropic inclusions
?2 exp(/|-.r)
sgn(*3) log-
R+\x-,
х\хъ
^3 exp(/|-x)d|
tan"
J 1
2 , V2 | ('
2 Al "ГЛ3
where R = (x? + x2 + jc32I/2.
The expression A4.1) can be reduced to
n= 1
where
00 / exp( /| • c)
d|. A4.5)
Integrations with respect to | lead to
-772<Sgn(c3) log"
R+\c,
C1C3
A4.6)

* 14. Cuboidal inclusions
107
Inclusion Molrl
Cur
Д
D
С
___
ve o,/of
5
t
1/5
- ellipsoif
—
05 i.o is го
к,/о, o.
О5 10 15 20 '
Fig 14 2 Variations of normal stresses along the Xj-axis for three different values of a1/a2,
where a2 = a3 (Dotted lines denote stress values for ellipsoidal inclusions )
where с = (c1; c2, c3) and other components are obtained by the cyclic permu-
permutation of A, 2, 3).
It can be seen that there are logarithmic singularities in certain edges and
corners of the cuboid, depending upon the types of eigenstrains in the cuboid;
i e., the singularities only occur in shear stress components when e* has no
shear strain components. ,
The stress components are obtained from о^(х) = Cijkl{uk t{x) — е^Ддг)}.
Figures 14.2 ~ 14.4 are taken from Chiu A977). Poisson's ratio v = 0.3 is used
in all calculations. For simplicity it is assumed that all components of the
Fig 14.3 Variations of normal stresses along the x2-axis for a1/a1 = 5 and a2 = a3.

108 Chap. 2 Isotropic inclusions
eigenstrain, except efj, vanish and that a2 is equal to a3. Figure 14.2 shows
variations of the normal stresses along the л:j-axis for the three types of
cuboidal domains. The curves show that the normal stress components are
continuous across the boundary of the cuboid at xx = av The dotted lines are
the stress components in an ellipsoidal inclusion with the semi-axes ax, a2,
a3 = a2. Figures 14.3 and 14.4 show that a22 is continuous across the boundary
of the cuboid at x2 = a2, while the other two normal components of the stress
are discontinuous at x2 = a2. It can be seen from F.11) that the discontinui-
discontinuities are local quantities which are independent of the overall shape of the
inclusion (independent of ax/a2 in the present calculation). It is found that
[<ju] = 1.099?^? and [a33] = 0.329?^?, where E is Young's modulus.
Lee and Johnson A977) have performed an alternative calculation by using
A1.30) and A1.31). The ф(х) and ф(х) defined by A1.35) have been obtained
by MacMillan A930) when Q is a cuboidal domain. They are
Y A4-7)
*(*)= L(-i)"?(cJ,
where
D(c) = \cxc2c3R + i{{R2 - c2)c2c3 log(R + q)
+ {R2- с22)съсх \og{R + c2) + {R2 - c2)Clc2 log(R + c3)}
?(c) = cxc2 \og{R + c3) + c2c3 log(R + q) + съсл \og(R + c2)
-\-{c2 t&n-\c2c3/ClR) + c2 t<m-l(c3Cl/c2R)
+ cj t<m-\Clc2/c3R)},
where с = (c1; c2, c3), R = (x2 + x\ + x2)l/2, and cn is defined in A4.2). The
strain can be derived from A1.32) where %j = e* ф and Ф,у = t* ф.
Lee and Johnson A977) further have calculated the elastic strain energy W*

* 14. Cuboidal inclusions
109
04
02
0
-02
-04
-06
-08
-
1
-
о2» а,
а, /ог--1
/S^ 2 3 4
Fig 14.4. Vanations of normal stresses along the x2-axis for a cuboidal inclusion ал = a2 = a3
of a single cuboidal inclusion and the interaction energy between two cuboidal
inclusions, where c*, = €*j, and other components are zero.
From A3.3) and a,., calculated from A4.7) or A4.4), Lee and Johnson
A977) obtained the elastic strain energy for ax = a2 = «з/Д,
A4.9)
where
-4B -
/32)
2I/2}
v\l logC -
- 1)].
A4.10)
The elastic strain energy per unit volume of a precipitate is smaller for a
cuboid than for an ellipsoid when /8 > 0.35. This relation is reversed when
/3 < 0.35.

по
Chap. 2 Isotropic inclusions
Sankaran and Laird A976) employed another method of calculation. Start-
Starting from A1.1) we can write
Vi
Gtj(x-x')nkUS{x')
A4.11)
where Gauss' theorem is used, and where | Q | is the boundary of п, п is the
outward normal vector on the boundary, and G;y A = — (d/dx'k)G;j.
Sankaran and Laird have calculated the stress field around misfitting
precipitates in plate morphology, such as в' in Al-Cu and tj in Al-Au (see
Aaronson, Laird, and Kinsman 1970, Sankaran and Laird 1974). The stresses
are shown to attain very high values near the interfaces at the edges. This can
significantly influence the generation of dislocations to accommodate the
mismatch.
Argon A976) has obtained approximate solutions for the stresses around
slender finite elastic rods and platelets with moduli different from those of the
infinite matrix.
15. Inclusions in a half space
Green's functions
A semi-infinite domain is defined by x3 > 0 as shown in Fig. 15.1. The surface
x3 = 0 is free from external tractions. The objective is to express the elastic
field when the eigenstrain efj(x) is given in the half space.
Fig 151 Half-space with an ellipsoidal inclusion at с from the surface

15. Inclusions in a half space 111
Green's functions G,7(*, *') for the half space are defined by the following
equations: For x3 > 0,
CijklGkmJj(x, x') + 8im8(x, x')=0 A5.1)
and on x3 = 0,
CijklGkmJ(x, X')nj = 8im8s(x, x'), A5.2)
where Gkm Длг, x') = (d/dx,)Gkm(x, x') and n} is the unit normal on surface
x3 = 0. 8(x, x') and 8s(x, x') are the three- and two-dimensional Dirac delta
functions, respectively, having the following properties:
[ f(x')8(x,x')dx'=f(x) A5.3)
¦'o
and
ff(x')8s(x,x')dS(x')=f(x) atx3 = 0, A5.4)
Js
Js
where S is the plane x3 = 0 and 8(x, x') = 8s(x, x') = Q when хФх'.
G^ix, x') is the x, component of displacement at point д: when a unit force
(body force when x' is an interior point and surface force when *' is on S) in
the ^-direction is applied at point x'.
The displacement м,(дг) caused by «*,(*) is expressed in a form similar to
C.31),
q,^*,,(*')?7M*' *') dx'- A5-5)
It can be shown that A5.5) satisfies the equations of equilibrium B.12),
c,jkiukjj(x) = CiJkltti,j(x), A5.6)
and the boundary condition B.13),
Cijklukj(x)nj = Cijkli*kl(x)nj. A5.7)

\\2 Chap. 2 Isotropic inclusions
In view of the properties of Green's functions, we have
Cpqmn<i*mn{x') — GkpJj(x, x')dxr
= -/o Cpqmni*mn{x')—^p8(x, x') dx'
= -J cij
< x') dx'
= f"cijklt*klJx')8(x, x') dx' = Cljk/e*k/Jx) A5.8)
and on the plane л:3 = О
= J
^mn(x')^rGkptl{x, x')nj dx'
Лх' x')nj dx'
where 5" is the surface x3 = 0.
A5.9)
Green's functions GtJ(x, x') for the semi-infinite isotropic medium have
been found by Mindlin A953); they are
R2 ij
R\
+
Ri
в.-.--
R\
4A -v)(l- 2v)
= GJx,x'),

75. Inclusions in a half space
G3Ax,x')= У J j) !<
G,3(x,x') =
16 TTJLt A — v)
(>x3x'3{x3 + x'3
|
R\
113
A5.10)
3-.
-2хгхъ
where
[f
R\ = {x, -x[f + (x2 - x'2f + (x3 + x'3)\
n5
R2
(x3-x'3)
A5.11)
ju is the shear modulus, and v is Poisson's ratio.
In these equations, the effect of the free surface appears in terms which
contain R2. (A minor sign error in the original expressions has been corrected
in a private communication; Mindlin 1975.)
The inclusion problem here occurs when e* is given in domain Q. If e* is a
constant, A5.5) becomes
Cjlmn(
* [~G (x x')dx'
'O OX, J
JU oxi
A5.12)
or
,.,(x, x')n,{x') dS(x'),
A5.13)
where Gauss' theorem is used, and | ?2 \ is the surface of fi.

114 Chap. 2 Isotropic inclusions
Recently, N.G. Lee A979) and Pan and Chou A979) have obtained explicit
expressions for Green's functions in a half space of a hexagonal continuum
with its basal plane as the free surface. Pan and Chou A979) further extended
their solutions to the two-phase transversely isotropic materials.
Ellipsoidal inclusion with a uniform dilatational eigenstrain
The elastic field is calculated when an ellipsoidal inclusion with a uniform
dilatational eigenstrain
is located near the surface of a half space. When A5.10) and A5.14) are
substituted into A5.12) we obtain
-2x3( —I |dx', A5.15)
where
and
x'l
a\
the domain
x'\ (x
a22
If we set
of
a
integration is
-0'
2 ~ '
3
Х\=У\> х2=у2, x3=y3 + c, x[=y{, x'2=y^, x'3=y^ + c A5.16)
and
хг =zx, x2 = z2, x3=z3-c, x[ =z{, x'2 = z'2, x'3= -z'3 + c, A5.17)

15. Inclusions in a half space
115
/"•
i *--—«..»¦•
Fig. 15.2a. Coordinate systems for Л, integral. Fig. 15.2b. Coordinate systems for R2 integral.
then Rx and R2 (see Fig. 15.2) become
„л2
,'\2
.'\2
A5.18)
J?j is the distance between points x and x'. R2 is the distance between point
л: and the mirror image of x'. In order to use formulae A2.9) ~ A2.13) in
A5.15), domain ?2 is transformed into its mirror image &2г when the integral ja
dx'/R2 is considered (this domain transformation is not necessary for /fi
dx'/RJ. Referring to A2.9) and A2.13), we write
A5.19)
where
U=\-
and we write
Уз
— = 1 for ex tenor points д: of 12,
X = 0 for interior points x of 42,
az^i
A5.20)
A5.20.1)

116
where
U=l-
Chap. 2 Isotropic inclusions
a\ +s
A5.20.2)
= 1 for any point x.
Then A5.15) reduces to
1
— v)
and, using A1.39), we obtain
A5.21)
The stress components are
A5.22)
A5.23)
where e* = 0 for exterior points of ?2. For interior points, the stress compo-
components become
°iAx) =
A5.24)

75. Inclusions in a half space
and for exterior points, they became
117
*.¦;<
A5.25)
Mindlin and Cheng A950) have treated the same problem, except that their
inclusion is a sphere (аг = a2 = a3).
The following numerical calculations are taken from the paper by Seo and
Mura A979), with v = 0.3 for all cases.
First consider the case when the inclusion with a1 = a2 = 3a3 is located at
depth с = a3. The dimensionless stress distributions are shown in Figs. 15.3 ~
15.4 by solid curves. For comparison, the results for a spherical inclusion
(a1 = a2 = a3) are shown by dotted curves. Figure 15.3 shows au, ст33 compo-
components along the x3-axis. For the ellipsoidal inclusion, ou is compressive at
interior points and increases with x3. It changes discontinuously to tension for
i.o
0.5
a.
CM
-0.5
¦a, «So,
Fig. 15.3. Stress distributions on A-A section.

118
Chap. 2 Jsotropic inclusions
-0.5
Fig. 15.4. Stress distributions on B-B section.
exterior points and decays to zero at infinity. The spherical inclusion, however,
causes tension for the interior points near the free surface. The stress compo-
component <j33 is continuous and takes a maximum tension at an interior point. The
maximum compression appears at the interface point. Its magnitude is quite
different for the two cases, and the compression region is broader in the
ellipsoidal case than in the spherical. To see the effect of the free boundary
surface, the stress distributions for an infinite medium are shown in Fig. 15.5.
It can be seen that the stress uniformity inside the inclusion for the infinite
medium is diminished by the existence of the free surface and some region
near the surface undergoes tension. Note also that component a33 in the
matrix may be positive (tension in regions) far away from the inclusion, while
a33 for the infinite medium is compressive everywhere.
In Fig. 15.4 the stress distributions on the B-B section are shown. The stress
jump at the interface is observed as expected for each component of the stress.
The effect of the free boundary surface appears in the a33 component showing
a tension region around the center of the ellipsoid. The effect of the inclusion
shape on the stress distribution can be seen by comparing the interface point
xl/a2 — 1 for the sphere with xl/a3 = 3 for the ellipsoid: au = a22 = o33 for
the sphere, but an = a22 Ф a33 for the ellipsoid.
The effect of depth с is shown by curves in Figs. 15.6 ~ 15.7. au is tensile
at the boundary surface and decreases with x3. It jumps to compression for

/5. Inclusions in a half space
119
0.5 -
Z -05 ~
-
1
\
\
1.0
/
У
--
го,----
— а.
-- а,
4.0
¦0,
¦о.
А
2о
«S
А
* За5
•о, •
-|
?\«
1
а
«,/а3
-€.О '
Fig. 15.5. Stress distributions on A-A section in case of infinite solid.
interior points of the inclusion and changes to tension again in the matrix.
Due to the existence of the free surface, the compressive stress components for
interior points of the inclusion increase with depth from the surface. This
effect is less prevalent for the spherical inclusion. It may be said that the effect
of the free surface disappears for the spherical inclusion, when Ola.
0.5
0
-0.5
V,,
А
t
—
—
2.0
-?!'_
а.
о,
3
' Q 3
——.
-
1
/
/
1
1
1
« а2 ¦ За
¦о.-о,
А
го,
А
s
¦ а
=Г
60
«1
^Г^5
-4
Fig. 15.6. Stress distributions on A-A section.

120
Chap. 2 Isotropic inclusions
10 2.0 30 4.0 5.0 6.0 7 0 8.0
Depth of the Ellipsoidal Inclusion, с/a.
Fig. 15 7 Effect of the inclusion depth on the surface stress
In Fig. 15.7 the effect of с on the stress component au at xY = x2 = x3 = 0
is shown for various aspect ratios of the ellipsoid. For the spherical inclusion,
au shows large tension at the surface that decreases with с It becomes almost
zero when с > За. аи at the surface decreases when the ellipsoid becomes
flatter. However, the effect of с is less prevalent in the flatter inclusion. It is
seen that the eigenstress at the surface depends not only on the depth of the
inclusion from the surface but also on the shape of the inclusion.
The elastic strain energy is, from A3.3),
A5.25.1)
Substituting from A5.24), we have
A5.25.2)
where V is the volume of the inclusion. The factor 2/t(l + v)(e*JV/(l - v) is

75. Inclusions in a half space 121
the elastic strain energy A3.21) for the case of an infinite space. The second
term in A5.25.2) is the correction factor due to the free surface.
If the inclusion is a sphere, we have
3>-= V/r, A5.25.3)
where r = (zf 4- z\ + zjI//2. Since
/6с\ A5.25.4)
,33
the correction term becomes — A + v)a3/6c3. Eshelby A961) has given this
correction term as — A + >»)a3/4c3 which involves a trivial numerical error.
The force acting on the inclusion is defined as
A5.25.5)
Thus, the free surface attracts the inclusion.
Tsuchida and Mura A983) solved the case when the inclusion is spheroidal
and the eigenstrain is symmetric about the axis of the spheroid.
* Cuboidal inclusion with uniform eigenstrains
When a cuboidal inclusion with a uniform eigenstrain is located in a half-space,
the elastic field can be obtained from A5.12) ~ A5.13) by taking ?2 as the
cuboidal domain. The surface integrals (on \Q\) in A5.13) have been ex-
explicitly performed by Lin and Tung A962) in their calculation of the stress
field produced by a localized plastic slip at a free surface. Chiu A978, 1980)
has extended his solution for an infinite body to the case of a half-space by
applying on the boundary plane equal and opposite normal and shear stresses
in order to obtain the free boundary conditions. Lee and Hsu A985) have
calculated the thermal stress when the cuboidal domain peeps out at the free
surface.
* Periodic distribution of eigenstrains
When eigenstrain e* (x) is given by
<f,(*)=<7>mexp(/?-*) A5.26)
in a half-space, the displacement field is the sum of C.10) and u*(x), where

122 Chap 2 Isotropic inclusions
и*(х) is the displacement caused by applying on the boundary plane x3 = 0
equal and opposite normal and shear stresses which would render this boundary
stress-free. Since the stress field for the full space problem is C.12), the xt
component of the force to be applied on the boundary x3 = 0 is
-^}^ exp(i{-x°)
= fjexp(i{-x°), A5.27)
where x° is a point on the boundary, ?• x° = |1x1° + ?2*2> and n is tne
outward unit normal on the boundary. If Ft is a unit force, the induced
displacement field is given by A5.10) with x'3 = 0. Thus
«f(x) = /Г Ffi^x, x°) exp(,'€.x°) dxfdx». A5.28)
' "'-00
The stress field of the problem is the sum of C.12) and afj, where
-x^d^dx». A5.29)
Owen and Mura A967) have performed the above integrations and obtained
o*;(x) = Щ^-^[Нктехр{;и1Х1+ ?2x2) - ?x3}], A5.30)
where
€-(*? + ЙI/а,
Hkm = V-l[8km-ix3{83kZm + 83Jk-83k83MK + iiM + iZ)} A5.31)
The solution in this subsection has many applications when the Fourier
series and integral methods are employed. When eigenstrains are given by

15. Inclusions in a half space
123
Fig. 15.8. Joined half spaces.
summations or integrations of A5.26), the corresponding solutions can be
constructed from summations or integrations (with respect to |) of the
solution given here. Owen and Mura have calculated the stress field of a Frank
dislocation network located near a free surface. Owen A971) has further
calculated by the present method the stress disturbance near the free surface
of a fiber-reinforced material subjected to tensile traction. Saito, Bozkurt, and
Mura A972) have considered the case of a periodic distribution of dislocations
in a thin film.
Joined half spaces
Let us consider the case when an infinite body consists of two half-spaces with
different shear moduli and Poisson's ratios (see Fig. 15.8). When one of the
half-spaces, e.g. region 1, contains an inclusion 12 with eigenstrain tfj, the
displacement field can be expressed in a form similar to A5.5),
J -
Нтп тп\ / 1}7 ^i /у ' ' '
ОХ/
A5.32)
It is easily proved that A5.32) satisfies A5.6); this is done in the manner
outlined in A5.8). Since t*mn is zero outside of ?2,
с, х') djc'.
A5.33)

124 Chap. 2 Isotropic inclusions
(?,у(дг, x') in the above expressions is Green's function in the two semi-infinite
solids; it is the x,-direction displacement at point x caused by the j^-direction
unit body force applied at point x'.
The elastic moduli in regions 1 and 2 are denoted by subscripts 1 and 2,
respectively.
When the two half-spaces are perfectly bonded, continuity of all compo-
components of displacements and tractions at the interface is required. The Green's
function for this case has been found by Rongved A955). The solution is given
in terms of the Papkovich-Neuber displacement functions (see Sokolnikoff
1956),
Gu(x, x') = (±G){(k + 1)Фи- {хкФк; + ^) J, A5.34)
where к = 3 — 4p and point x' is in region 1. For x in region 1,
2w(k + 1
-2Ax'3(Xj-x;)/R\},
+ (AK1-S)x'3(xJ-xr)/R2(R2 + x3 + x'3)}, A5.35)
= Ф2з = О,

15. Inclusions in a half space 125
For x in region 2,
Ф21 = Ф12 = О,
-в-т)
ф, = -
+1) /?!</?! -X3+X'3)' ,„-д.
A5.36)
- Л - Г )
ф
33
where
<\ 7 = 1,2,
[f+ (x -x'f
Л? = (^ -x[f+ (x2 -x'2f + (х3 - х'2)\
R\ = (Xl - x[f + (х2 - x'2f + (х3 + дсО2,
А = A - Г)/A + К1Г), ? = (к2 - кгГ)/(к2 + Г),
S = A - Г)/A + Г), Г= 2(Kl + 1)Г/(к2 + 1)A + Г).

126 Chap. 2 Isotropic inclusions
When the two half-spaces have smooth contact, the boundary conditions
for Green's function are continuity of the normal components of displacement
and stress and a condition for vanishing shear components of stress at the
interface. The Green function for this case has been found by Dundurs and
Hetenyi A965). For л: in region 1, (x' is in region 1),
«i - !)(*; - x'j)/2(R2
+ x'3(xj- x'j)/R2(R2
Ф13 = Ф23 = О,
For jc in region 2,
*11 = Ф22 = *12 = Ф21 = О,

15. Inclusions in a half space Yll
-*;
A5.39)
where
г, 7 =
A5.40)
Special cases Г = 0, 1 and 00 correspond to a single half-space, a homoge-
homogeneous whole space, and the smooth rigid base contact, respectively.
It should be noted from A5.35) ~ A5.40) that A5.34) can be denved from
Green's function for a homogeneous infinite medium. The terms containing
R2 and R1-x3 + x'3 in A5.35) ~ A5.39) can be constructed from i?j by
proper elementary operations (changing arguments, multiplying by (Xj — х'у),
differentiating, and integrating). Aderogba A977) has pointed out that an
elastic solution for dissimilar joined half-spaces is constructed from the
solution for a homogeneous whole space.
Dundurs and Guell A965) and Guell and Dundurs A967) (see also Aderogba
1972) have calculated the stress field for a center of dilatation where
e*(x)=6,7e*6(x1, x2, хъ-с). A5.41)
From A5.33) this case gives
",(*) = C^AjpG^x, x')\ A5.42)
[aXl \xi-0,x'2 = 0,x', = c
According to Dundurs and Guell, the normal traction transmitted by the
interface is

128 Chap. 2 Isotropic inclusions
for the smooth interface, and it is
for the perfectly bonded interface. It is interesting to compare the above
results with the solution for the whole space (a center of dilatation is placed at
point @, 0, c)),
(i5-45)
Then we have
(азз)*,-o = 2^03*3 for smooth surface,
A5.46)
= A - A) (X3*3 for perfect bond.
Moon and Pao A967) have calculated the interaction energy between a
spherical inhomogeneity and a center of dilatation. The center of dilatation
may be attracted to or repelled by the inhomogeneity, depending on the
rigidity of the inhomogeneity and the matrix material. Recently, Zhang and
Chou A985) have obtained explicit expressions for the stress field and strain
energy for an antiplane eigenstrain problem of an elliptic inclusion in a
two-phase anisotropic medium.

Chap. 3 Anisotropic inclusions 129
Anisotropic inclusions
In this chapter we investigate elastic fields due to ellipsoidal inclusions in
anisotropic materials. Since explicit expressions for anisotropic Green's func-
functions are not available, the technique used in the last chapter is not applicable.
We start from the result in Section 6 and discuss how to integrate the integrals
involved in the Fourier space and physical space.
The results are presented in general forms, which are applicable to any
distribution of eigenstrains given in an ellipsoidal inclusion. Known solutions
can be derived as special cases from these general results.
16. Elastic field of an ellipsoidal inclusion
When eigenstrain efy-(x) is distributed in an ellipsoidal sub-domain Q of an
infinitely extended anisotropic material, the displacement is expressed by F.2),
OXIJS2
-(x-x')}d€ A6.1)
or, changing ? to - ?, by
и,(х) = -B^Г3^- °°
axl
Xexp{-/?-(x-x')} d?, A6.2)
where the integral with respect to x'v x'2, х'ъ is defined in domain i2, which is
the ellipsoid
xl/al + x\/a\ + x\/a\ < 1
(see Fig. 11.1).

130 Chap. 3 Anisotropic inclusions
The integration with respect to the |-space is considered first. The volume
element in the ?-space, d?, is
d|=d^d|2d^3 = |2d|dS(|), A6.3)
where
and dS(?) is a surface element on the unit sphere S2 in the |-space. Then,
A6.1) is written as
ox i
X«p{i€-(x-x')}d5(«), A6.5)
and A6.2) leads to
6xlJi2 J-x JS2
Xexp{it-(x-x')}dS(i), A6.6)
where | is changed to — ?. The sum of A6.5) and A6.6) is
«,(*)- -iB-)~3^/ dW" dif CJlmne*nm(x')NuU)D~\0
Xexp{/«-(x-x')}dS(O. A6-7)
In the above integrals the homogeneity of degree zero of ?2JV^.(?)D~4?) nas
been used, namely,
, ^4#)- A6-8)
Since j™x exp(/|rj) d| = 2w8(tj), A6.7) becomes
A6.9)

16. Elastic field of an ellipsoidal inclusion
131
Fig 16 1 Unit sphere and parameters
The following transformations of variables are used:
X\/al=yl, x2/a2=y2, x3/a3=y3,
x'i/ai=y[, x'2/a2=y'2, x'3/a3=y'3,
«i?i = Si, «2S2 = S2> «з^з = S3.
A6.10)
We have also
djc' = dx[ dx'2 dx'3 = ala2a3 Ay[ dy2 dy3 = axa2a3r dr d<|> dz, A6.11)
where г, ф and z are shown in Fig. 16.1 and
z = t~-y' = i-x'/S. A6.12)
Note that A6.10) transforms the ellipsoid into the unit sphere S2. Since
Э/Эх,в(?• (x - *')) = €,«'(«¦ (* - *')). A6-13)
where 5' is the derivative of 8 with respect to the argument, A6.9) is written as
-I •'O •'O
A6.14)

132
where
R = {l-z2f
must be a real number.
Differentiation of A6.14) with respect to x} leads to
Chap. 3 Anisotropic inclusions
A6.15)
-I
A6.16)
The argument f(f У ~ z) of the derivatives of Dirac's delta function becomes
zero when
f-j> = z. A6.17)
When point у is inside the unit sphere, z2 < 1 for any f, and R becomes a real
number. When point у is outside the unit sphere, R is real for only certain f.
In other words, for x e ?2, the integration domain for J is the whole surface of
the unit sphere S2, and for д: <? Q the integration domain becomes the
subspace of S2 satisfying the condition z2 < 1. The subspace will be denoted
by S*, which is the unshaded domain seen in Fig. 16.2.
The expressions A6.14) and A6.16) and the other modified expressions
appearing in later sections remain in the same form for any arbitrarily
oriented coordinate system, except that ? is taken as
A6.18)
Fig 16 2 S* is a subspace of S satisfying condition |f у | < 1 for a given у

17. Formulae for interior points 133
where a,, is the direction cosine between the Jty-axis and the Ith principal
direction of the ellipsoid.
17. Formulae for interior points
Let us consider the case when point л: is inside the ellipsoid (interior point) or
point у is inside the unit sphere. Since S'(ff у - ?z) = — ?~1(d/dz)8(H?-y —
fz), the integration of A6.14) by parts with respect to z leads to
-i Jo
where R and x' are only functions of z for a fixed ? through A6.15) and
A6.12). The boundary values of the integration by parts vanish, since
«(?f-J + r)=0 A7.2)
for interior points. Furthermore, for the interior points there exists a z
satisfying the condition
?-y-z = 0, -l<z<l, A7.3)
for any f. Then, A7.1) can be integrated with respect to z,
, A7.4)
where
*={l-z2}1/2. A7.5)
The points x' in A7.4) must satisfy the condition
!(S-y-S-y')=i-(x-x') = o A7.6)

134
Chap. 3 Anisotropic inclusions
or
= S-y. (П.7)
Similarly, A6.16) becomes, after twice integrating by parts with respect to z,
«„;(*)=-
a\a2a3 f2m
A7.8)
where x' is subject to condition A7.6), and R is defined by_A7.5). The above
result can also be derived by differentiating A7.4), where (S-y)j = I"/ and
R,j = -(f-j)^/"^ at z = (f • y). It can be seen from A7.8)'that if e*m is
constant, Uj j is also constant, and if e*m is a linear function of coordinates,
м. у is also a linear function of coordinates.
The surface element dS(?) can be transformed into a new surface element
defined at point f by the use of A6.10). Let dS(?) be a surface element
constructed by the vector product of d? and S?, and dS(f), by the vector
product of df and 5f, where dfx = ax d{x/l, 8? = a^^/f, etc. (see Fig. 17.1).
Then,
«2
A7.9)
Therefore, we have the relation
A7.10)
Uniform eigenstrains
The displacement and strain fields for interior points become as follows when

/ 7. Formulae for interior points
\У.
is uniform:
Fig 17 1 Transformation of the surface element
A7.11)
and
A7.12)
or, using A7.10),
/ dS(S)
A7.13)
and
A7.14)
Equation A7.11) has been derived by Kinoshita and Mura A971).
Furthermore, the surface element dS(?) can be written as
dfl,
A7.15)
where 0 is measured counter-clockwise from the ?\-axis as shown in Fig. 17.2.
Then, A7.13) is written as
1 J0
A7.16)

136
Chap. 3 Anisotropic inclusions
Fig 17 2 New coordinate system ?3 and в
where
G,jkl{i) = iJiNuU)/D($). A7.17)
If the notation introduced in A1.15) is used, we can write
? =S ?* , A7.18)
where
Sljmn =
From A6.10)
A7.19)
A7.20)
however, since Gljkl(^v |2, ^3) are homogeneous polynomials of degree 0, we
can use
A7.21)
for I as the argument of Gijkl(%). The components of f are
A7.22)
see Fig. 17.2.

17. Formulae for interior points 137
The integrals with respect to в in A7.16) or A7.19) can be obtained by the
residue calculation in a complex Z-plane, where
cos в = {Z+Z~l)/2,
sin6l = (Z-Z-1)/2i, A7.23)
dd = dZ/iZ.
Then,
* = 2vR[Gijkl/Z], A7.24)
where R[Gijkl/Z] is the sum of the residues of the function GiJkl/Z existing
within the unit circle | Z | = 1 under a fixed value of f3. Then A7.16) is
written as
«,.(*) = j^CJlmif.*nmxkGtjkl, A7.25)
where
jl[\ df3. A7.26)
j[
Also, we have, from A7.19),
Sun.n = (V^)Cpqmn(G,Pjq + GJpiq). A7.27)
Some special cases are considered below; see Lin and Mura A973).
Spheroid (a1 = a2, al/ a3 = p)
The crystalline directions are assumed to be coincident with the principal
directions of the spheroid.
For cubic crystals, nonzero components of Gijkl are given below:
111 = 222
- flA, *)Л/?A - ^2 + P2x2) + yp2x2} dx,

138 Chap. 3 Anisotropic inclusions
- _477 /4PV* 2 2 2U 2h_ 2
a Jo pq y /L
77Y /"I P2^2(l - X2) ,
' dx,
e •'o P(P + <1)
. = G2211
= — ГA~л2){A - x2 + p2x2)[m2(i - x2 + pV) + ppV]
a Jo pq
+ A - x2)[p(l - x2 + p2x2) + yp2x2} } dx
m /-i(l-x2Jr , x2 | 2;c2)+ v,
° ^ P * A7.28)
= ^2233
= — f^ {2A - x2 + p2x2)[M2(l - x2 + p2x2) + flP2x2}
-x2 + p2x2)+YP2x2]} dx,
313 ~ ^2323
A + MJ r1!
a i0
A + M) rip2x2(l-
a Jo p{p +
?)
^-^
2a

17 Formulae for interior points 139
where
p = a1/aJ,
(П.29)
p = {A - x2 + p2x2f + bp2x2(l - x2){\ -x2 + p2x2)
q = {A - x2 + p2x2f + bp2x2{\ - x2){\ -x2 + Рйх2)}1/2, 0 < x < 1.
It has been verified that p2 and q2 are positive for most of the cubic crystals.
When p = 1, GU11 and G1212 agree with Kneer's results A965). However, Gu22
of Kneer's is incorrect, since it does not reduce to the isotropic case.
For hexagonal crystals, the elastic moduli are denoted by
Cn=d,
г\Сп — Cl2) = e,
C44=f, A7.30)
C13 + Q4 = g,

140 Chap. 3 Anisotropic inclusions
where C;J are the Voigt constants. The nonzero components of Gijkl are given
below:
-g2P2x2(l-x2)} dx,
G3333
333 = 4тг f lAp2x2 [dil - x2) +fp2x2] [e(l - x2) +fp2x2} dx,
- х2) + 4fp2x2] - 3g2p2x2(l - x2)} dx,
A7.31)
= G2233 = 2^rAp2x2{[(d+e)(l-x2) + 2fp2x2}
x[f(l-x2)+hp2x2] -g2p2x2(l-x2)} dx,
= ^3322
= 277 (lA{\ - x2)[d{l - x2) +fp2x2} [e(l - x2) +fp2x2] dx,
•41
/2(l - x2)[e(l - x2) +fpV] dx,
o
where
A'1 = [e(l -x2) +fP2x2} {[d(l -x2) +fp2x2] [/A - x2) + hp2x2]
-g2p2x2{\-x2)}. A7.31.1)
When p = 1, all integrals except G3311 agree with Kneer's results A965).
Isotropic materials are obtained by a limiting process: u?-> X + 2ц, е -»ц, f
—* ju., g -> X + jx, and h -* X + 2ju.

17. Formulae for interior points 141
Cylinder (a3—> ooj (elliptic inclusion)
A rod or a needle-shaped inclusion can be approximated by a cylindrical
inclusion. When a3 -» oo, A7.21) becomes
? = ?/0!, #2 = f2A2, |3 = 0. A7.32)
SinceG ,,lc,(^i/a1, $2/a2, 0) are homogeneous functions of degree 0, the factor
A - Гз) /2 in A7.22) can be dropped. Then Gjjkl are independent of f3 and
we have
и,(х) = (l/2v)Cjlmne*mnxkfo °Gijkl{Uai, Уаг, 0) d0 A7.33)
with
fi = cosfl, f2 = sin0. A7.34)
When the coordinate axes xt are not taken along the directions of the
principal axes of the cylinder, we have
u,(x) = (l/2ir)CJlmne*mnxkf2nGijkl{ai/p/ap, alplp/ap, 0) dO, A7.35)
where p = 1,2 and aJp are the direction cosines between the x^-axis and the
pih principal directions of the ellipsoid.
If the cylinder has a circular cross section (ax = a2), Gijkl are obtained
from A7.28) by taking p = 0.
Jaswon and Bhargava A961) and List A969) were able to obtain explicit
solutions for interior and exterior points by using complex representation.
Jaswon and Bhargava's analysis has been extended by Willis A964) to the
anisotropic case with cubic symmetry. Subsequent anisotropic analyses have
been performed by Bhargava and Radhakrishna A963, 1964) for an ortho-
tropic medium and by Chen A967) for a medium with one plane of symmetry.
These treatments of two-dimensional inclusion problems are restricted to the
state of plane strain or plane stress, based on the classical formulation for the
plane theory of elasticity (e.g., Green and Zerna 1954). However, for a general
anisotropic medium it is not always possible to treat the plane strain and
antiplane strain fields independently. Yang and Chou A976) have presented a
general method for solving the generalized plane problems of inclusions in
anisotropic solids. They have used Green's function E.84) for the generalized
plane problems developed by Eshelby, Read, and Shockley A953).

142 Chap. 3 Anisotropic inclusions
The following is the result obtained by Yang and Chou A976) for ortho-
rhombic materials with elastic moduli given by (A2.17), where eigenstrain e*
is assumed to be uniform in the elliptic domain with the half-axes ax and a2.
For interior points,
_ _ л у Cl2 C12 j ^
12 CnQsm a 12>
where
j _ (с /С I/4 С =(С С I/2 е = а /а
a = cos(/:rC/2) for-4<C<0,
= cos(//C/2) for C> 0, A7.35.2)
С= (С12 + С12)(С12 - Си - 2С66)/С12С66,
Q = 1 + е2А2 + 2еА sin a.
The component a33 сап be obtained from the condition e33 = 0. When isotropy
is approached (X = 1 and C — 0), equations A7.35.1) reduce to the results of
Jaswon and Bhargava.
The strain energy is
W* =
u
e*22. A7.35.3)
C12Qsina l1'
Yang and Chou A977) further obtained solutions of antiplane strain
problems of an elliptic inclusion in an anisotropic medium.

17. Formulae for interior points 143
Flat ellipsoid (a3^> 0)
When a3 -> 0, from A7.21),
ii, iz^S,; A7.36)
therefore, we can put ?x = ?2 = 0 for the argument of GiJkl(?). Then
Gijkl@, 0, ?3) are homogeneous functions of degree 0, and we set i3 = l.
Therefore, A7.16) becomes
«,(*) = ^/m/L^G,7W@, 0, 1). A7.37)
When the coordinate axes are not taken along the principal directions of the
flat ellipsoid, we have
M;U) = CjimAmxkGijki{au, a23, a33), A7.37.1)
where aJp is the direction cosine between the Xj-axis and the pth pnncipal
direction of the flat ellipsoid.
When the x-coordinate axes are taken in the pnncipal directions of the flat
ellipsoid, the displacement in the flat ellipsoid is identical to zero, since
G^O, 0, 1) in A7.37) becomes Sk3Sl3Nu@,0, l)/D@, 0, I) and, therefore,
xk8k3 = x3 = 0 on the flat ellipsoid. Then we have
M, = o> ",,i = 0 and M,,2 = 0- A7.38)
The stress becomes
% = Cpqi3Cj3mn^mNu{i)D-^) - Cpqik<L*kl, A7.39)
where | = @, 0, 1). By definition, we have from E.5)
CpqJqZkN,M)D~l(Z) = V A7.40)
Since Ij = 0, | = 0, and €3 = 1, A7.40) is identical to
Then, A7.39) leads to
0,3 = 0, A7.42)

144 Chap. 3 Anisotropic inclusions
which should be expected for the case of a very flat inclusion. Furthermore, we
have from A7.37) and A7.41)
W/ „,\д + с*,.. A7.43)
If
??i = <$2 = «?2 = 0, A7.44)
it is shown that
H"l,3 + M3,l)=«13' H,3 + U3,2)=?23> ,3 = €?3 A7-45)
and any stress component vanishes,
% = 0. A7.46)
In short, when e*x = e*2 = f*2 = 0, all the stress components become zero,
resulting in vanishing elastic energy. Further, an arbitrary line element parallel
to the flat surface does not change, since ut x = ul ¦ 2 = 0 from A7.37).
If the coordinate system is taken arbitrarily, A7.39) can be written as
% ^^^f - Cpqi?*ki, A7.47)
where l"=(a13, a23, a33).
Eigenstrains with polynomial variation
For this case the stresses at interior and exterior points are investigated
simultaneously; see Section 19.
Eigenstrains with a periodic form
The elastic field inside an ellipsoidal inclusion is considered when the distribu-
distribution of eigenstrains inside the inclusion is periodic,
<:«(*') = CL exp(iC/,Jc;/fl,) = i*m exp(/C;,;/), A7.48)
where l*nm are constants, с is a given vector, and the summation convention
defined in A1.39) is used. From Fig. 16.1 we write
y' = zl + r cos фт + r sin фи, A7.49)

17 Formulae for interior points 145
and, therefore,
СрУ'Р = (c • f> + rco «*(ф - a), A7.50)
where (see Fig. 17.3)
(c-/w)/co = cos a, (c-n)/co = sin a, c0 = [c-c- (c-fJ} . A7.51)
Then we have
A7.52)
Substituting A7.52) into A7.4), we obtain
0 2 /
Й7Г •'O
Xexp{ /(f • J»)(c • f) + irc0 cos(^ - a)}
+ iRc0 cos((|) — a)}
/¦ C^MOD-KOU-^Sd). A7.53)
From the properties of the Bessel functions,
"ехр{//-с0со5(ф-а)} d<b = 2vJ0(cQr), {zJx{z)}' = zJQ(z), A7.54)
and A7.53) is written as
A7.55)

146 Chap. 3 Anisotropic inclusions
Fig 17 3 Illustration of vector с
Similarly, A7.8) is written as
A7.56)
where
«={1-(?-^J}1/2, S-y-i-хГ1- A7-57)
The result A7.55) has been obtained by Mura, Mori and Kato A976). This
solution can be called the fundamental solution. When e*m is given as an
arbitrary function, it can be expressed by the Fourier series or integrals in
terms of A7.48). The corresponding solution ut j{x) is the sum or integral of
A7.56) with respect to various values of c.
When the ellipsoid degenerates into a flat ellipsoid (a3 -» 0), expression
A7.55) simplifies. First the unit sphere S2 is transformed into a cylinder as
shown in Fig. 17.4, and the cylindrical surface element dS reduces to йв d|3
which is related to dS by
?f)V2. A7.58)

17. Formulae for interior points 147
Also, from A6.10),
ix = cos 0/A + i\f\ i2 = sin 0/A + ^I/2, i3 = «3/(l + g)l/\ 5
Г3 dS(t) = (a2 cos20 + a\ sin20 + ef«|)/2 d0 d?3,
where - oo < ?3 < oo.
With use of the identity
2 • 2л, 2t2\-V2
2 3^3 /
= (a!2cos20 + a^ sin20) 1^-|^3(a12 cos20 +a2 sin20 + a2^2) 1/:
A7.60)
integration by parts with respect to ?3 is applied to A7.55). Then we have
-(аха2аъ/*тт)Ск1тп1*птB* йв Г дф? cos20 + a\
'0 •'-oo
0 •'-oo
a2 sin20+ а2ъ
X
рр}Щ A7.61)
where A6.10) has been used.

148
Chap. 3 Anisotropic inclusions
Fig 17 4 The unit sphere S2 is transformed into an infinitely long cylinder
For ?3-»oo, from A7.59),
6 = 0, «2-0, #з = 1;
therefore,
(x-?)=x3, f = a3,
A7.62)
A7.63)
Furthermore, for аъ -* 0, the second integral in the right-hand side of A7.61)
vanishes. The factor [ ] in the integrand in the first integral is independent of
в and
f2'\a\ cos20 + a\ sin20) X
Equation A7.61) becomes, for аг -> О,
«.-(*) = Cklmni*nmx3J0{c0R)
A7.64)
A7.65)
where f= @, 0, 1).
Since we are considering a point л: inside the inclusion, we set x3 = 0 before

* 18. Formulae for exterior points 149
taking the limit a3 -»0. Then, A7.65) leads to
w,(x)=0 A7.66)
for interior points. By taking the derivative of A7.65) with respect to xJt we
obtain the expressions for the displacement gradient and, therefore, the stress
components throughout the inclusion,
A7.67)
-l*j exp{i(c1xl/a1 + c2x2/a2)}\,
where ? is the unit normal to the surface of the flat ellipsoid.
* 18. Formulae for exterior points
When point jc is outside an ellipsoidal inclusion (point у is outside S2), the
boundary terms produced by the integration by parts with respect to z in
A6.14) and A6.16) are not necessarily zero.
Since
A8.1)
A6.14) with A7.10) can be written as
(Rrdrf
A8.2)
J_l JQ JQ Jgl

150 Chap. 3 Anisotropic inclusions
Integration by parts with respect to z leads to
'2"
lim(8772)/'2
z -»1 ' 0
- lim i**2)-1}2" d4>f\drf
z-*-l J0 J0 JS
-1 •'0
-I
eaf-^-fz) dS(f). A8.3)
The first two integrals in A8.3) become zero, since i?-> 0 for z-» +1 and
Js2 &№'У~ ?z)dS(?) is finite. The integration with respect to z in the
remaining integrals is performed explicitly by the use of the property of
Dirac's delta function,
f -Sz)dz = S-y, A8.4)
-l •'-i
if | f у | < 1; otherwise, they are zero. Therefore, A8.3) becomes
z = $ у
A8.5)

* 18. Formulae for exterior points
where S* is a subdomain of S2 satisfying the condition
151
A8.5.1)
S* is shown in Fig. 16.2 by the unshaded surface on S2. The notations defined
in A6.10) and Fig. 16.1 are used in A8.5), and R = A - z2I/2.
Similarly, after repeating the integration by parts with respect to z, Mura
and Cheng A977) rewrote A6.16) as
ЪMil («.б)
The line integrals in A8.6) are defined on L+ in Fig. 18.1 where l-y = l. The
surface integrals on S* in A8.6) can be obtained directly from A8.5) by
differentiation with respect to xy 3/3xy = \?~х Ь/Ъг from A6.12).
Fig 18 1 L+ is the boundary of S* in the side of у

152
Chap. 3 Anisotropic inclusions
The above result can be denved directly by differentiating A8.5) with
respect to x.. Since R j= ~(f •J>)lJ/Kf and zy = ^/f at z = f y, the differ-
differentiation gives
A8.6.1)
The last term comes from the change in S* when x changes by 8x. It is clear
that this term is considered only along L+ and L~, where L is the circle
Fig 18 2 New coordinate system f'

* 18. Formulae for exterior points
153
Fig 18 3 Vectors in A8 6 3)
f у = -1. From Figs. 18.2 and 18.3,
3xy S7 ^L +
A8.6.2)
along L+. To understand this equation we note, in view of Fig. 18.3,
dS = -sin a d0 da,
sin a = l/y,
8y?=(8x ¦?)/?.
A8.6.3)
The last relation is obtained from A6.10). Similarly, 3S/3xy along L is given
by
ds
Thus, A8.6.1) is written as
A8.6.4)
r-«
г-fj

154 Chap. 3 Anisotropic inclusions
*cklmj N
-(*•>){«:.,(*')},-*[
*-Sy=-i
when z -» +1, Л -^0 and ^021Г аф -» 2тг. Furthermore, note from A7.49)_that
[«:«(*0],—i forjf]^ on_L" is equal to [e*m(*')]z=J for [f]L+ (= -tf]?-)
on L+, and Njk(?)D х(€)^ is an even function of ?. Thus, one obtains
Г"=Л
which is identical to A8.6).
If point x is inside the inclusion, the line integrals in A8.6) can be dropped,
and S* in the surface integrals is replaced by S2. The resulting equation is
identical to A7.8).

* 18. Formulae for exterior points 155
We note that the stress field has a discontinuity across the boundary of the
inclusion. When л: approaches the inclusion boundary from exterior points,
the integral domain S* in A8.6) becomes S2, and f, satisfying the condition
f у = 1, approaches y; that is, aJj/? -> xx/av a2f2/f ->x2/a2, and а31зД
-» x3/a3. It is easily shown that ?; —> и., where ni is the outward normal on
the inclusion boundary. The integrals along L+ in A8.6) are reduced to
~CkimneZm(x)Nik(n)D~l(n)nJnl. Thus, the stress jump on the inclusion
boundary is
[a,7]=a,7(out)-ao(in)
= Cijkl{-Cpqmn<L*nm{x)Nkp{n)D-\n)nqnl + i*l{x)} A8.7)
which agrees with F.11).
The stress concentration of an inclusion or inhomogeneity is obtained from
ao(out) which can be evaluated from A8.7) when a,7(in) and [a^] are known
(see Laws 1977). The stress concentration factor of a lens-shaped void and its
relation to the stress intensity factor of a crack (a3 -»0) have been discussed
by Mura and Cheng A977). The dynamic stress concentration factor of a
spherical cavity was investigated by Moon and Pao A967).
The integrals on S* in A8.6) can be further modified as follows: A new
coordinate system ?' is introduced as shown in Fig. 18.2, where the f^-axis is
taken in the direction of y, and the f{- and f2-axes are lying on the plane
perpendicular to y. Then we have
dS(f) = d? d$, A8.8)
where в is measured counter-clockwise from the fj-axis, and, therefore,
[ ] dfl, A8.9)
since S* is determined from ?-y = $'ъу = +1.
The integrands in A8.6) are functions of | and therefore must be expressed
in terms of f3 and в. The necessary transformation is as follows (see A6.10)):

156 Chap. 3 Anisotropic inclusions
Й = A-ГзI/2со8 0, A8.10)
where atJ is the direction cosine between the ?- and ^'-axes.
* Examples
It will now be shown that A8.5) and A8.6) lead to well-known solutions. Let
us consider a sphere with radius a and with a uniform dilatational distribution
of the eigenstrain,
<*=«,/*• A8-11)
Consider A8.5) with A8.9) at point (xv 0, 0). Then, the ?-axis coincides with
the fj-axis, and ?, = ?.. Equation A8.5) becomes
«i(*i, 0, 0) = %Cklmn^ AOJa/Xl GlkU(() d«-, A8.12)
where Gljk,a) = М(ДЩ
For an isotropic material, we have
\*
Because of the symmetry of the problem, the above result can be written as
и = a3e*(l + v)/3r2(l — v), where и is the displacement in the radial direc-
direction. Then er = Зм/Зг = -2a3e*(l + v)/3r3(l - v), which agrees with A1.45).
Next, consider the case when a centrally symmetric eigenstrain is prescribed
in a sphere,
«•(*) = «,.,«• | x|. A8.14)

* 18. Formulae for exterior points 157
Then, the use of the notations in Fig. 16.1 and A6.10) leads to
* 3 1 - A8.15)
r?c;m(x')dr =a8nmc*a^)(l-|rj;0-
Consider point (xv 0, 0). Equation A8.5) yields, for an isotropic medium,
Ax}
or
a\* \ + v
^ (!«•«)
и =
4r2 -1 -
A8.17)
which is identical to и obtained from A1.43) by taking aT~c*r. Equation
A8.6) becomes
0 a/xi
or
3u 1 1 + v a4f*
2 1-
A8.19)
It can also be shown that A8.6) leads to A1.33) when e*j is uniform and the
material is isotropic. This result agrees with Eshelby's solution A959).
In cases where analytical results are obtained from A8.5), equation A8.6)
does not necessarily have to be used for uir as seen in the considered
examples. However, A8.6) is necessary for numerical calculations when no
analytical expression is available from A8.5).

158 Chap. 3 Anisotropic inclusions
19. Ellipsoidal inclusions with polynomial eigenstrains in anisotropic media
We consider the case when eigenstrains inside the inclusion are given in the
following polynomial form of x' or y':
<*пт(*') = 'пМГШа2ЫГ, A9-1)
where enm are constants and ах, а2 and «3 are integers. As shown in Fig. 16.1,
we can write
y; = ^p + r(mp cos ф + пр sin ф), A9.2)
where m, n, f are orthogonal unit vectors. The binomial theorem leads to
U)ei= E RU "l! R,Az^r-^r^m, cos ф + п, sin ф)^. A9.3)
The integrals with respect to ф in A8.5) have the form
/ (m1 cos ф + nx sin ф) (m2 cos ф + п2 sin ф) 2
x(m3 cos ф + п3 sin ф)ръ йф. A9.4)
A complex number Z = exp(/<?) is introduced here. By defining
Wp = \{mp + inp), Wp = \(mp-inp), A9.5)
where / = J — l, we can write A9.4) as
x L —r^—r-w^
<?2 =

19. Ellipsoidal inclusions with polynomial eigenstrains 159
where \Z\ = 1 is a unit circle in the complex plane. The residue calculation is
applied. The constant terms in the product of the power series of Z give the
residues. Then, A9.6) is written as
**? /V ft! ft!
X W^-qiWi2-q2W^~q'W^W^W^. A9.7)
The integral A9.6) becomes zero when \{PX + /?2 + /?3) is not an integer; that
is, A9.6) reduces to A9.7) when jSx + уб2 + /?3 is even; otherwise it is zero.
Therefore equation A8.5), after integration with respect to r, becomes
a2 3 /
\ Р (^ + 2)/2 а -
E ш-jO (i-(f-j'))
X
(F,
A9.8)
where a = ax + a2 + a3, /3 = /3X + jS2 -Ь jS3 and the summation with respect to
/?!, /J2, /?3 is taken only when /J is even. The д: dependency of w, can be seen
from terms containing (?-y) = (| •*)/?. If a equals N, then м, is the sum of
the polynomials of degrees N + 1, N —I, N —2,..., for interior points, since
S* = S2 for these points.
The displacement gradient can be obtained as the sum of the direct
derivative of the integrand of A9.8) and the line integral on L+, as seen in
A8.6), where
{<т(х')},-1=<птё?&&> A9-9)

160 Chap. 3 Anisotropic inclusions
since in Fig. 16.1 y' -* f when z -> 1. Equation A9.8) has been given by Mura
and Kinoshita A978).
Asaro and Barnett A975) have obtained essentially the same expression.
The elastic field for exterior points of the inclusion becomes important
when the interaction between two or more inclusions is considered. An
approximation method has been proposed by Moschovidis and Mura A975)
for two isotropic inhomogeneous inclusions. The elastic field in the second
inclusion caused by the first inclusion is approximated by its Taylor's expan-
expansion at the center of the second inclusion. The sum of the two elastic fields
caused by individual inclusions can satisfy the required conditions for the
equivalent inclusion problem at the second inclusion. A similar procedure is
applied for points located in the first inclusion.
Special cases
When a given eigenstrain is linear,
we have from A9.2)
and, therefore,
A9.12)
When
Cn(*') = CUJ, A913)
we have, similarly,
A9-14)

* 20. Harmonic eigenstrains 161
When the ellipsoid degenerates into a flat ellipsoid (a3 -»0), an argument
similar to A7.36) can be employed. For interior points we have, from A9.12),
A9.15)
where ? = ^/av /2 = t?2/a2, |3 = ff3/a_3, and yt=xx/ax, y2 = x2/a2, y3_
= хъ/аъ. Gikjt(iiY, ?2, ?3) = Arjfc(|)Z)~1(^)^^( is a homogeneous function of ?
of degree zero and independent of the geometry of the elUpsoid. When a3 —> 0,
?i> ?~2 ^ ?~з>il can be written as Gikjl{Q, 0, 1). The integral in A9.15), therefore,
can be reduced to the integral
A9.16)
where V is the volume of S2 and np is the normal vector on S2. Finally,
A9.15) is written as
utJ{x) = CklmnePmypGikjl@, 0, 1). A9.17)
Similarly, we have, from A9.14),
",,,(*) = Cklmymny2Glkjl@, 0, 1), A9.18)
which is the displacement gradient inside a flat ellipsoid when an eigenstrain is
given by A9.13).
* 20. Harmonic eigenstrains
Let us consider the case where the eigenstrain component ?*, is a solid
harmonic function of у in an ellipsoidal inclusion fi,
€*j(x) = ?*j{b})ynPn( у ¦ w/y), B0.1)
where yl = x1/av уг = х2/а2, уъ = хъ/аъ, and у = {у(у,I/2\ w is an arbi-
arbitrary vector on the unit sphere S2; and Pn is the Legendre polynomial of
degree n. It is known that any function /(y) of class C1 defined on S1 can be
expanded in a uniformly convergent series,
/Ы= Е -^i-//(«)Pn(^-W)d5(W) B0.2)

162 Chap. 3 Anisotropic inclusions
Fig. 20.1. Vectors in B0.3).
(see e.g., Hobson 1931). Therefore, the eigenstrain given by B0.1) may have a
broad application, particularly in an interaction problem of two ellipsoidal
inhomogeneities subjected to an applied stress. Note that B0.1) is not a solid
harmonic function of x.
When B0.1) is substituted into A6.14), we have
Ky'yp^y' ¦ a/y')8'(g-y - U) dS(S), B0.3)
where y[ = x[/aY, / = (KtfI/2, ?=*?/?, ? = (<#"? + аЩ + аЩГ\
and z, ф, г and R are shown in Fig. 20.1. We have
y' = zi + r(m cos ф + п sin ф). B0.4)
The addition theorem of the Legendre polynomial leads to
р„{ у' ¦«//) = РЛ y' ¦ Ш)рЛ- «)
+ 2 E fe^C4 / -Vy')P^'«) со<тф). B0.5)
Since /02" cos w</> d<#> = 0, only the first term in B0.5) is used for Pn in B0.3).
Furthermore, we have
2" йф[*гйг(у'УРя(у'.[/у>)
^r dr(z2 + r2)n/2Pn(z/iJz2 + r2)
= 2w A - z2)P;+1(z)/(n + l)(/i + 2) B0.6)

* 20. Harmonic eigenstrains 163
and
= f1 - (и + 1)(и + 2)Рп + 1{2)ГЩа-У - U)
•'-I
= 0 for lf-j-1 >1. B0.7)
Therefore, B0.3) can be written as
B0.8)
where S* is the subspace of S2 satisfying \$ -y\ < 1. For interior points of S2,
S* is taken as S2. When и = 0, then Po = 1> pi = f •J' = € ^Г1, and B0.8)
becomes identical to A7.11).
The above result, B0.8), can be further modified, since Nik(^)D~~l(^)^ are
homogeneous functions of degree — 1. Using A7.10), we have
„.(*) = A/4*)/" Qni{L u,)Pn + 1(!-y)dS(n, B0.9)
Js*
where
Qjl *) = Cklpf*pq{«>)Nik{Z)D-\l)ZlPn{i-<*),
<2ni can be expanded in a series of surface harmonic functions,
B0.11)

164 Chap. 3 Anisotropic inclusions
The above expression is substituted into B0.9) and the integration on S* is
performed. The addition theorem,
Т^'Л^ -У) cos(/fl),
yields
f Pm(S-ri)Pn + 1(S-y)dS(S) = 24rPm(ii-yHnm(y), B0.12)
Js*
f
Js*
where y=y/y, y=\y\, d5(f) = йв At, t = $-y, and
B0.13)
for interior points x of ?2, since S* = S2, and
-l/y У J -1
for exterior points дг, since S* is determined from the condition
\S-y\ = \ty\<l. B0.15)
The integrals B0.13) and B0.14) can be performed by the use of Rodrigues'
formula (Whittaker and Watson, 1962),
^«) = ^?>-1Г. B0.16)
By repeated integration by parts, B0.13) becomes
i + l-m)/2
\ Z /
? \ ? / „m + 21
1=0
B0.17)

21. Periodic distribution of spherical inclusions 165
where у < 1, m < n + 1, and Г is the Gamma-function. When m > n + 1,
Фпш = 0. Similarly, B0.14) becomes
( -n'rf 2w~2/+1
Ф„
/=o /!r( w - и - 2/ + 1 \ r/ w + и - 2/ + 4
B0.18)
where у > 1 and n + 1 < т. When n + 1> m, Фпт = О. In addition,
= ^T form = « + l. B0.19)
Finally, B0.9) can be written as
00 r\ _|_ -i
и,(х) = 5 X) —-г Фпт(у) ( Qni(v, w)Pm(v-p) dS(rj). B0.20)
From the definition B0.10), terms for which n + m is even, are zero. It should
be noticed that B0.20) is a uniformly convergent series and the integral
domain S is independent of лг or y. Therefore, the displacement gradient ut ¦
can be obtained directly from B0.20) by differentiation, where ду/дхг =
Ух/а-\У and 9(i] • у)/дхх = щ/а\^ е*с.
21. Periodic distribution of spherical inclusions
Let us consider a periodic distribution of spherical inclusions with a period 2L
in all three directions in the Cartesian coordinates (Fig. 21.1(a)). The distribu-
distribution of eigenstrains can be expressed in the Fourier series as
OO
•*), B1.1)
where ?x = -nvx/L, |2 = ttv2/L, ?3 = ttv^/L (excluding vv = v2 = рг = 0, since
the uniform eigenstrain in a whole space gives a divergent solution), and
<* U) = JjjIJt'U*) exp(i€- x) dx. B1.2)

166
Chap. 3 Anisotropic inclusions
(a)
(b)
1С)
Fig 21 1 Periodic distributions of spherical inclusions in a unit cell
Similarly, the displacement w,(x) can also be expressed in the Fourier series
form
B1.3)
According to C.14), the solution can Ъе written as
«,(*)--* ETT'
= " 00
B1.4)
If e* (jc) is uniform inside the spherical domains and zero outside, then
B1.2) becomes
B1.5)

21. Periodic distribution of spherical inclusions 167
The integral has been carried out by Asaro A975) and gives
e*= (а/2Ь)\2пУ/2^-3/%/2Ы' B1-6)
where a is the radius of the sphere,
and
h/i D) = BАЧ )V2( 4 -x sin r, - cos т,). B1.8)
Finally, the displacement field ut(x) becomes
и (x)=-i(—\V С ?* VVV w mn-iffU „-3/2
"i\XJ '-)/¦) 4lmn€m.
If the spherical inclusions are distributed in a body-centered cubic lattice (see
Fig. 21.1(b)), the solution can be the sum of B1.9) and modified B1.9) where
jc is replaced by л: - с. If the spherical inclusions are distributed in a
face-centered cubic lattice (see Fig. 21.1(c)), the solution is obtained as the
sum of four expressions similar to B1.9) having exponential factors exp(z? • x),
exp{i?-(x -cj}, exp{/?-(x- c2)} and exp{/'?•(.* — c3)}. The distortions
corresponding to these cases can be summarized by
3/2
/()(v)exp(i?-x) B1.10)
where
{1 \ for simple cubic
l + exp(-i{-c) I forb.c.c.
) + exp(-(| ¦ c2) + exp(-i?¦ c3)) for f.c.c.
B1.11)

168 Chap. 3 Anisotropic inclusions
The elastic strain energy assigned to a single inclusion is obtained from
A3.3) as
W* = -\ f a, ?*, d?>, B1.12)
where Vo is the volume of a single inclusion, i.e., Fo = Ana3/3. aiy in Fo can be
calculated from B1.10) and B.4). Then, B1.12) becomes
X ГЕН' Gkplq{v)v-%2/2{^)F(v), B1.13)
where F(v)is, the real part of f{v).
Khachaturyan A969, 1970) and Khachaturyan and Airapetyan A974) have
studied a spatial distribution of the inclusions by a different method and
concluded that the periodic distribution proves to be stable.
In some alloys, clusters or precipitates are formed in a periodic manner
(Ardell, Nicholson, and Eshelby, 1966; and Eurin, Penisson, and Bornett,
1973), which is usually interpreted in terms of the interaction of two neighbor-
neighboring clusters or precipitates. However, the elastic strain energy in such a
situation can be directly calculated from B1.13). Mori et al. A978) have
calculated the elastic strain energy for periodic arrangements of nitrogen
atoms in iron. It is assumed that a nitrogen atom occupies a spherical region,
Fo, the volume of which is equal to that of an iron atom and has eigenstrains
of ?*3 = 0.86 and e^ = t*22 = -0.08 (Nowick and Berry 1975, misfitting sphere
model). There are numerous ways of arranging the nitrogen atoms in periodic
arrays. Mori et al. A978) have calculated three cases: simple cubic, body-
centered cubic, and face-centered cubic arrangements, all of which have
symmetry identical to that of the host iron atoms. The elastic constants are
assumed to be Cmi = 2.37 X 10n N/m2, and Cm2 = 1.41 X 10u N/m2,
c\i\i = 1-16 X 10n N/m2. The calculated energy is plotted against 2L in Fig.
21.2. First, it is seen that the lowest elastic strain energy is attained when the
nitrogen atoms are arranged in body-centered cubic arrays. Secondly, in the
simple and face-centered cubic arrays, the elastic strain energy decreases
monotonically to the asymptotic value as the period 2L increases. The
difference between the asymptotic value and the elastic strain energy in any
specified arrangement is the interaction energy among all the nitrogen atoms.
Contrary to the cases of the simple cubic and face-centered cubic arrays the

21. Periodic distribution of spherical inclusions
169
в
"О
1
36
35
34
33
\
\
\
\
\
—^.^
\
\
\
\
\
\ "¦
\
\
\
^^Face-Cen
^\ с
Simple Cubi
/'Body Cer
Cubic
tered
ubic
\
с
. — ¦
tered
1 2 3 4 5 6 7
2L/a
Fig 21 2 Elastic energy per nitrogen atom
elastic strain energy in the body-centered cubic array first decreases and then
increases toward the asymptotic value as the separation increases. The lowest
energy state is attained when 2L/a = 4.0. It is noted that this situation
corresponds approximately to the arrangement of nitrogen atoms in the a"
precipitate, Fe16N2, observed as an intermediate phase in supersaturated
iron-nitrogen alloys. Although one cannot examine all the possible arrange-
arrangements of the nitrogen atoms, it is clear from the above results that the
formation of the a" intermediate phase on tempering of supersaturated
iron-nitrogen alloys can be understood on the basis of the elastic energy
associated with the clustering of nitrogen atoms.
Instead of taking the approach described above, it is possible to use a direct
method which utilizes the summation of the interaction energy between
nitrogen atoms arranged in a periodic manner. Although Brown et al. A973)
reported that the summation converges very slowly, Furuhashi, Kinoshita, and
Mura A981) recently have proved that the summation of the stress fields of
individual inclusions does not converge.
If a bounded inclusion ?2P with eigenstrain ej? is given in an infinite
three-dimensional space D, the stress a/J due to the inclusion can be obtained
from F.1).
If an infinite number of inclusions пр (/> = 1, 2, ...) are distributed in D,
the stress atj may be the sum of oft,
a (x)= У ар(х) B1-14)
The convergency of this series, however, is not obvious. Furuhashi, Kinoshita,

170 Chap. 3 Anisotropic inclusions
and Mura A981) showed that the convergency depends on the dimensions of
the distributions (line, plane, or three-dimensional) and the types of eigen-
strains (uniform or linear function of coordinates, etc.). The summation
B1.14) is replaced by a new summation when it does not converge.
Let us assume that the material is isotropic and e*f = Stj for all indices p.
According to Timoshenko and Goodier A951), о?(огр or of) has the form
J+(x2J+(x3J}3/2 B1.15)
for exterior points of Qp, where xv x2, хъ are the Cartesian coordinates
measured from the center of spherical inclusion Qp.
The following Eisenstein's theorem (see Whittaker and Watson, 1962) is
used:
Il/(?12 + ?22+...+?B2)' B1.16)
in which the summation extends over all positive and negative integral values
and zero values of qx, q2,...,qn, (except the set of simultaneous zero values),
is absolutely convergent if s > \n and divergent if s < \n.
Suppose that the distribution of Qp is defined in the region хг > 0, x2> 0,
x3 > 0 as
p{; |p|},
B1.17)
P = (Pi, Pi, Рз)> Pi> Pi' Рз'- positive integers.
In order to investigate the convergency of B1.14), it is sufficient to investigate
the convergency of the series
5= E l/{(x1-3PlJ + (x2-3p2J+(x3-3p3J}V2. B1.18)
Pl-PZ'Pl
Since it is always possible for a given set of xt and pt to choose the smallest
integers qt such that
\xl-3Pi\<qi, B1.19)
then, for such qt,
S> E 3/2 )

21. Periodic distribution of spherical inclusions 171
Since n = 3, s = f in Eisenstein's theorem, the series on the right-hand side in
B1.20) is divergent and, therefore, series B1.18) diverges.
Although qx, q2, q3 in the series B1.20) do not take all integers used for the
series B1.16), Eisenstein's theorem still holds. Consider the series X^ >„_1/| p
- yx |3 instead of B1.18), where n. are arbitrary positive integers such that
pt — \xt > 1 for pt > и., and let qt be the smallest integers such as pt — \xt < qr
Then the inequality
+ q\ + q\)m B1.21)
holds for some positive integers q'r
It is concluded that the resulting stress caused by a three-dimensional
periodic distribution of spherical inclusions with uniform dilatational eigen-
strains cannot be expressed as B1.14). The expression for the stress field must
be modified, as will be shown later.
Let us consider the following plane distribution of inclusions:
«,= {*; \х-Ър\<\),
B1.22)
P = (Pi> P2'®)> Pi> Pi- a^ positive and negative integers.
In order to investigate the convergency of B1.14) it is sufficient to investi-
investigate the convergency of the series
S= I l/{(x1-3p1f + (x2~3p2J + xi}3/2. B1.23)
РъРг
It is always possible to choose the largest integers qv q2 for given values of xt
and pt such that
\х1-Ър1\ >qlt \x2-3p2\ >q2. B1.24)
Then
Ч22У/2. B1.25)
When Eisenstein's theorem is applied to the right-hand side in B1.25), the
series converges because n = 2 and j = f. An argument similar to that follow-
following B1.20) can also be applied if one wants strictness of argument on the

172 Chap. 3 Anisotropic inclusions
convergency. It is concluded that for any plane or linear penodic distribution
of inclusions, the resulting stress can be expressed as B1.14).
Now let us construct a suitable expression for the resulting stress caused by
a three-dimensional periodic distribution of inclusions. In order to have a
convergent series, we consider
B1-26)
Since
B1.27)
where M and С are finite constants, the series B1.26) is convergent (n = 3,
s = |). An arbitrary constant can be added to B1.26) when B1.14) is replaced
by B1.26). The arbitrary constant must be determined from the condition of
periodicity.
Let us define v0 as a unit cell of the periodic distribution of inclusions. For
distribution B1.17), v0 is a hexahedron defined by — § < x, < \, i = 1, 2, 3.
The integration otj over v0 becomes
/ аи(х) dx = CiJk!f { ukJ(x) - ?*,(*)} dx
 
= CIJklj uk{x)n, dS - CiJkJ ??,(*) dx, B1.28)
where | v01 is the boundary of v0 and Cijk, are the elastic moduli. The
condition of periodicity gives
f uk(x)nidS = Q. B1.29)
•Vol
Then, B1.28) becomes
faiJ(x)dx=-Cljk/J4l(x)dx. B1.30)
«о Щ
J

21. Periodic distribution of spherical inclusions 173
In order to satisfy the above condition, the stress expression must have the
form
j (< < «„, B1-31)
p
where
«„= -{\/vo)[c,jkljet,{x)dx+ j I(a?(x)-a?@))dx). B1.32)
Expression B1.14) can be recovered from B1.32) when the summation B1.14)
is convergent. It can also be observed that expression B1.31) is independent of
the choice of the origin of coordinates.
To generalize the last result for eigenstrains prescribed by polynomials of
the coordinates, we start our discussion from equation A2.2). The displace-
displacement field for interior and exterior points of Q is expressed as
^Г7){^,-4A-,)Ф,/,/}, B1.33)
where
*„.(х) = /|х-х'|??,-(х')<1х' B1.34)
and
1^ BL35)
Фи(х) and ^(x) are the harmonic and biharmonic potentials due to a body
О of density «*¦(*'); they have the properties described by A2.5) and A2.6).
Let €* (x) be any homogeneous polynomial of degree /,
<%(х)=Вф1 ,х,х1г...ха. B1.36)
Then, the structure of the elastic field can be determined by the property of
the integral
^1*2 s' JB \x-x'\ V '

174 Chap 3 Anisotropic inclusions
According to Dyson A891), when 12 is a sphere with radius a, the integral
B1.37) for exterior point x of п is explicitly written as
[1/2]
B1.38)
where [1/2] is Gauss' symbol (a maximum integer not exceeding 1/2) and
C(n a) = — ^^ Г ( '" r Г21 39)
C{"'a> 22"«!(« + l)! h (e + n-Z-r-i)- ^^
The finite series B1.38) can be summarized in the form
{/o(*)/l*r+/i(*)/l*r~2+-+/(' + 2)/2(*)}/l*l/+1 B1-40)
if / is even, and in the form
}l/+2 B1-41)
if / is odd, where ff are homogeneous polynomials of x of degree / for all
indices j. Accordingly, the displacement expression B1.33) can be summarized
in the form
()/}+5 B1.42)
if / is even, and in the form
{ 1/+6 B1-43)
if / is odd.
The corresponding expression for stress becomes
{/о;+4(^)/1^|/+2+/1'+4(^)/1^1'+..-+/('Д42)/2(^)}/^1'+7 B1-44)
if / is even, and
if / is odd.

21. Periodic distribution of spherical inclusions 175
If an infinite number of inclusions are periodically distributed in the
three-dimensional space as shown by B1.17), the resulting stress can be
obtained by the sum of B1.43) or B1.44), depending on /. These summations
are taken for all Qp (p = 1, 2, ...). The coordinates appearing in B1.44) and
B1.45) are measured from the center of each Qp. Convergency of the resulting
series can be investigated by Eisenstein's theorem. The theory is applied to the
last terms in B1.42) ~ B1.45), since they have the lowest 0(l/|x|). Since
n = 3 and s = \ for B1.45), the series constructed from B1.45) is convergent.
It is concluded, therefore, that the stress field can be expressed as B1.14) when
the eigenstrain is a homogeneous polynomial with degree of an odd number /.
The other series representing displacement fields and the stress field for an
even number of / are generally divergent. However, the series may converge
when the last few terms in B1.42) ~ B1.44) are zero. It is always possible,
however, to write the stress field in the form of B1.31).
We show that expression B1.31) with B1.32) is valid even for arbitrarily
inclusions in an anisotropic medium with an arbitrary eigenstrain, where e*f
is integrable in Qp. i2p are identical for all p.
For Green's function of elasticity Gsk(x), we have E.61),
Gsk<rl(x) = V(x)/\x\\ B1.46)
where V(x) is analytic for x Ф 0 and positively homogeneous of degree 0. Then
V(x+p)= V(p) + V i(p + 0x)xi,{0<e<\) B1.47)
and
. . уг,\ , , i ¦ , -¦ B1.48)
х+р\3 \х\3 УГ'\\х+р\3 \x\3J \x+p\3
From B1.27) the series constructed from the first term in the right-hand side
in B1.48) converges. Furthermore,
|F,(x)|<j^ B1.49)
and
^T7T^T^L^M^'(l/'l>|jr|)- BL50)

176 Chap 3 Anisotropic inclusions
Therefore, the series constructed from the last term in the right hand side in
B1.48) is also convergent. From F.1) we have
B1.51)
where e* „ is integrable in Q ; therefore, the series in B1.31) and B1.32) are
convergent.

Chap. 4 Ellipsoidal inhomogeneities 177
Ellipsoidal inhomogeneities
When the elastic moduli of an ellipsoidal subdomain of a material differ from
those of the remainder (matrix), the subdomain is called an ellipsoidal
inhomogeneity. Voids, cracks and precipitates are examples of the inhomo-
geneity which might also be called an inclusion. However, the term "inclusion"
has been used in a different context in this book. Here, an inclusion has the
same elastic moduli as the matrix, and it contains eigenstrains.
A material containing inhomogeneities is free from any stress field unless a
load is applied. On the other hand, a material containing inclusions is
subjected to an internal stress (eigenstress) field, even if it is free from all
external tractions.
If an inhomogeneity contains an eigenstrain, it is called an inhomogeneous
inclusion. Most of the precipitates in alloys and martensites in phase transfor-
transformation are inhomogeneous inclusions. Eigenstrains inside these inhomoge-
inhomogeneous inclusions are misfit and phase transformation strains.
Eshelby A957) first pointed out that the stress disturbance in an applied
stress due to the presence of an inhomogeneity can be simulated by an
eigenstress caused by an inclusion when the eigenstrain is chosen properly.
This equivalency will be called the equivalent inclusion method.
This book emphasizes the equivalent inclusion method, since it provides a
consistent method whereby the results of inclusions in homogeneous media
obtained in Chapters 2 and 3 are used. Exact methods of analysis for stress
concentration around ellipsoidal cavities in isotropic media subjected to a
uniform tension or torsion usually involve harmonic or bi-harmonic displace-
displacement potentials. Particular solutions of this kind have been developed by
Lame A866), Boussinesq A885), Galerkin A930), Love A927), Papkovich
A932), Neuber A934) and Mindlin A936). A brief survey of this approach has
been given by Sternberg A958). The solution for a spherical cavity under an
arbitrary uniform field at infinity is attributed to Neuber A937) and Leon
A908). It has also been given by Southwell and Gough A926), Goodier A933),
Sternberg, Eubanks, and Sadowsky A952), Sternberg and Sadowsky A952),
and Ling and Yang A951). Sadowsky and Sternberg A947), Shapiro A947),

178 Chap. 4 Ellipsoidal inhomogeneities
Miyamoto A950, 1951, 1953), Edwards A951) and Das A954) have given
solutions for a spheroidal cavity or inhomogeneity. For an ellipsoidal inhomo-
geneity or inclusion, solutions have been given by Sadowsky and Sternberg
A949), Lurie A952), Robinson A951) and Niesel A953). The problem of a
spheroidal inhomogeneity in a transversely isotropic material has been solved
by Bose A965) and Chen A966,1968). In connection with a geotechnical study,
Selvadurai A976, 1978) investigated the load-deflection characteristics of deep
rigid anchors embedded in cohesive soil or rock media. The displacement
potential functions used by Edwards A951) were slightly modified by Tsuchida
and Nakahara A974) so that various half-space problems with a cavity could
be solved (see, e.g.,Tsuchida and Saito 1980). Tsuchida and Mura A983)
obtained the stress field in a half-space having a spheroidal inhomogeneity
under tension.
22. Equivalent inclusion method
Consider an infinitely extended material with the elastic moduli CijkI, contain-
containing an ellipsoidal domain ?2 with the elastic moduli C*jkl. Q is called an
ellipsoidal inhomogeneity. We investigate the disturbance in an applied stress
caused by the presence of this inhomogeneity. Let us denote the applied stress
at infinity by a° and the corresponding strain by К"?,_/ + "",¦)• The stress
disturbance and the displacement disturbance are denoted by otj and м;,
respectively. The total stress (actual stress) is a° + atj, and the total displace-
displacement is м° + и,-. Stress components atJ are in self-equilibrium; that is,
a,,,. = 0 B2.1)
and atj = 0 at infinity. When a finite body is considered,
о„и,. = 0 B2.2)
on the boundary, where n( is the normal unit vector on the boundary.
Hooke's law is written as
°и + °и=С?.к1 «,+ «*,,) in О,
B2.3)
°° + °,, = <W<,+ «„,,) in D-Q.
The equivalent inclusion method is used to simulate the stress disturbance
using the eigenstress resulting from an inclusion which occupies the space Q.

22. Equivalent inclusion method 179
Consider an infinitely extended homogeneous material with the elastic
moduli Cijkl everywhere, containing domain п with an eigenstrain e*.. c*j has
been introduced here arbitrarily in order to simulate the inhomogeneity
problem by use of the inclusion method. Such an eigenstrain is called an
equivalent eigenstrain. When this homogeneous material is subjected to the
appUed strain с 9. = ^(u° . + u9(.) at infinity, the resulting total stress, distor-
distortion, and elastic distortion, respectively, are a,9. + atJ, u®j + ut Jt and м°у + ut}
— ?* in п. Then, Hooke's law yields
B2.4)
°?j + °,j = CuM.1 + «*./) in D-Q,
where e° = CiJk,u°kJ.
The necessary and sufficient condition for the equivalency of the stresses
and strains in the above two problems of inhomogeneity and inclusion is
с**,К, + "м) = с^К/+ "*,/-<*/) ini2 B2-5)
or
C*k/{4l + 4l) = C,JuK + 4,-4,) inO. B2.5.1)
As mentioned in the preceding sections, ek! in the above equation can be
obtained as a known function of i*kl when the eigenstrain problem in the
homogeneous material is solved. Thus, B2.5) determines e*, for a given (°kh in
such a manner that equivalency holds. After obtaining е?л the stress a,° + atJ
can be found from B2.3) or B2.4).
If a° is a uniform stress, c* is also uniform in 42. Then, from A1.15) or
A7.18)
«*/=!(«*,/+«/.*) = ««„/;„, B2.6)
where Sklmn are given by A1.16) for isotropic materials and by A7.27) for
anisotropic materials.
Substitution of B2.6) into B2.5.1) leads to
C*kl\€kl+ Sklmn€t,n) ~ ^ijklK^kl^ ^klmrftin ~ ?*/) B2.7)
from which the six unknowns, efj, are determined.
Sometimes the inhomogeneity may involve its own eigenstrain. Then, it is
called the inhomogeneous inclusion. In other words, a domain which has

180 Chap. 4 Ellipsoidal inhomogeneities
different elastic constants from those of the matrix may also have a given
distribution of eigenstrain efr An example is the formation of martensite
blades in quenched carbon steel and precipitations in alloys.
Let a material D, containing an ellipsoidal inhomogeneous inclusion Q, be
under a stress field a,° + aiy. a° is the applied stress if the material is
homogeneous, i.e. having no inclusions. atJ is the sum of the two stress
disturbances, one caused by the inhomogeneity, and the other, by the eigen-
stress associated with eigenstrain efj in ?2.
Denoting the elastic constants of D - fi by CiJkl and those of Q by C*kl,
we have Hooke's law,
B2.8)
where
a°,. = 0 in D,
o?jnrFl on\D\, B2.9)
aiJy = 0 in D,
The inhomogeneous inclusion is simulated by an inclusion in the homogeneous
material with eigenstrain efj plus equivalent eigenstrain e*y,
o°j + oij=CiJkl(ull+ukJ-ePk,-c*!) in 0,
B2.10)
°?j + au = СцМ.1 + uk,i) in D - 0.
Eigenstrain efj is a fictitious one, introduced for this simulation. The equiv-
equivalency between B2.8) and B2.10) holds when

22. Equivalent inclusion method 181
If a° is a given uniform stress field and efy is a given uniform eigenstrain in
Q, B2.11) is satisfied by taking €tJ as the solution of the inclusion problem
with a uniform eigenstrain ??¦ + ?*¦. Instead of B2.6), we now have
4i = i(«k>/+ «/.*) = ski>*n«n + О s 5,/m/**n, B2.12)
where
?**=?? + ?*. B2.12.1)
Substituting B2.12) into B2.11), we have the equation to determine ?**,
au
= Сок/(е2;+5к/и^я-?2Г). B2.13)
In the absence of the applied stress, this becomes
°V = C*kl(Sklmne*m*n - ej>,) = Cijkl(Sklmnt*m*n - c*f). B2.13.1)
Isotropic materials
Let us illustrate in more detail the equivalent inclusion method described
above by assuming the material to be isotropic. When both the matrix and
inhomogeneity are isotropic, B2.11) can be written as
,. + ?0.-??;)+X80(ck + ?At-?j?), B2.14)
where \*, \i* and X, ju are the Lame constants in the inhomogeneity and the
matrix, respectively, and
B2.15)
<f/= «?,- + <&.

182 Chap. 4 Ellipsoidal inhomogeneities
It is convenient to introduce the following deviatoric (reduced) strains:
'«й = «&-М?*/3>
B2.16)
'<Г; = ??;-80<Е:/з,
Then B2.14) is equivalent to
B2.17)
**(«** + «** - <b) = Ki*lk + 4k - €??),
where
*" = X + 2ju/3 = ?/3A - 2к),
? = 2A + i»V, B2.18)
\ =
and K, E, and у are the bulk modulus, Young's modulus, and Poisson's ratio
of the matrix, respectively. Similar relations hold among the elastic constants
of the inhomogeneity.
If ajj is a given uniform stress field and e^ is a given uniform strain in Q,
we can use B2.12) for solving c* from B2.17). The shear components of «**
are obtained directly from the first equation in B2.17):
С12
{2(/i-/i*)ei°2 + 2/i*e
!3 + 2ft}, B2.19)
The determination of efj*, ?** and ef3* is not a simple matter. After e** are
calculated, the stress field in 12 is obtained as
a° + a,., = 2M(e° + S,,mnc**n - с**) + Х8и(е°кк + Skkmnt*m*n - e**). B2.20)

22. Equivalent inclusion method
183
<=r°,
Fig 22 1 An ellipsoidal inhomogeneity under stress a°
The strain outside Q is given by A1.33) and A1.41).
Let us consider the case where applied stress a° is uniform at infinity (see
Fig. 22.1) and the inhomogeneity is an inhomogeneous inclusion with a
uniform plastic strain field efi, ef2, ef3. When a° and ifj are shear, the
solution has already been given by B2.19) and B2.20).
Sphere
From A1.15) and A1.21), e,7 inside the spherical inclusion with uniform
eigenstrams e^*, e?f and e^3* are given by
e*2*
15A - v)en = G - 5v)e*n* + Ep - l)e*
15A - v)i22 = Ev - l)c** + G - 5v)e** + Ev -
15A - v)e33 = Ey - l)cf* + Ev - l)e** + G - 5
and
Then,
B2.21)
B2.22)
B2.23)

184 Chap. 4 Ellipsoidal inhomogeneities
With the aid of this last result, it follows from the first equation in B2.17) that
'c** = 15 {'e° (ц* - м) - '</>*}(! - ")/{E" - 7)m - (8 - lOv)fi*}. B2.24)
The second equations in B2.17) and B2.22) yield
tti = 3{(K*-K)e°kk-K**Zk}(l-v)/{Dv-2)K-(l + v)K*}. B2.25)
Thus, we have
ef/ = 15{'e° (/i* - m) - '?/>*}(! - ")/{E" - 7)M - (8 - 10»0/i*}
+ 8ij{(K*-K)elk-K*ek>k}(l-v)/{Dv-2)K-(l + v)K*}.
B2.26)
The stress components in i2 can be calculated from B2.20). For the case when
tension о is applied in the x3-direction, 'е°г = 't\2 = -о/6ц, 'е°ъъ = о/Зц and
Penny shape
When i2 is a penny-shaped inhomogeneity, ax = a2 = a and a3 = 0. From
A1.23), ?,7 inside fi, for given c^*, ?*2* an^ ?*з*> are
ii22 '')+?22''/A-'')+4з*- B2-27)
Then, B2.14) becomes
- 2fi)e°u - 2/xVi + (X* - X)e2* - X*«ffc + X*€33
(X* - \)e°kk - \*^k + X*c33
-,), B2.28)

22. Equivalent inclusion method 185
It should be noted that a33 is zero. When B2.28) and B2.27) are solved for
efi*> ?** an°l €зз*> one obtains
v)e** = {Bц-2,1*)^ + 2М*е^} - к{Bм* -
2m*L + 2М*е?3}A - v)X*/(\
- 2,x)eou + 2M*efi}
- {Bц - 2^*)e°3 + 2р*€& }A - ^)Л*/(А* + 2^*)
+ {(X- Х*Укк + X*ePk}(l - vJp*/(\* + 2m*), B2.29)
2мA + v)e** = - {Bм - 2м*К°! + 2M
+ {Bм - 2м*)с?з + 2м
The stress components are calculated from B2.20).
Rod
A rod-shaped geometry results when a1 = a2 = a and a3 = oo. From A1.22),
e,y inside i2, for given cj^*, and €^2*, cj3*, are
8(! " ")<n = E - 4»-)e** + Dp - l)e** + 4и«,
8A - «/)е22 = Dy - 1)?** + E - 4v)t% + 4ve**, B2.30)
«33 = 0-

186 Chap. 4 Ellipsoidal inhomogeneities
Substitution of B2.30) into B2.14), after some modification, gives
= {Bft - 2М*
(cf2 - ?f3)}(l - p), B2.31)
(X*p
ef* are obtained from these linear algebraic equations.
If the problem is symmetric, so that «ft = ef2, e?x = e°2 anc^ ?** = ?**> we
have
B2.32)
where
J'2). B2.33)
It should be noted that the stress components are independent of the size of
the inhomogeneity, but are dependent on its shape.

22 Equivalent inclusion method
Anisotropic inhomogeneities in isotropic matrices
187
When an anisotropic inhomogeneous inclusion is contained in an isotropic
matrix, Cjjkl and Sklmn in B2.13.1) are taken for the isotropic material. The
calculation then follows that of the isotropic inhomogeneity case.
Stress fields for exterior points
Displacement and stress fields outside of an ellipsoidal inhomogeneity, J2, can
be obtained from the formulae in Sections 12 and 18 when the equivalent
eigenstrain is determined. However, these formulae are complicated in prac-
practice. Recently, Tanaka and Mura A982) proposed an alternative method to
evaluate the elastic fields for points exterior to J2.
Consider an ellipsoidal inhomogeneity Q with elastic moduli C*k! in a
matrix with elastic moduli Cijk, and an applied stress ст° at infinity. First,
obtain the stress field for the points interior to J2 and denote it by a/J + a^(in).
Next, find the stress field for the points exterior to J2 if Q is a void and the
applied stress is —a,, .(in). Denote the stress field by ajy(out). Then, the stress
field for the points exterior to fl in the original problem is the sum of
afj + au(in) and o0(out) (see Fig. 22.2). The proof is obvious. a,7(out) becomes
-a,.y.(in) at infinty and zero in J2. Therefore, a° . + a,7(in) + a,7(out) is a° at
infimty and аЯ + a, (in) in 12, which are the required conditions in the original
problem. This method is useful since solutions for the void problem under
various stress fields are usually available in the literature (e.g. Neuber 1937,
Muskhelishvili 1953, Savin 1961, Peterson 1962, Miyamoto 1967).
cr*
-07. (in)
СГ.-КХ. (in)
>! 'I
--ijM
Fig 22 2 The method for obtaining the exterior elastic field

188
Chap. 4 Ellipsoidal inhomogeneities
23. Numerical calculations
The equivalent inclusion method can be extended to cases where applied
stresses are not uniform. If an applied stress is linear with respect to the
coordinates, the equivalent eigenstrain is also linear with respect to these
coordinates. Generally, when the applied stress is a polynomial of degree n,
the equivalent eigenstrain is also chosen as a polynomial of degree n. This
arises from the fact that for a polynomial eigenstrain of degree n the
corresponding eigenstress of the inclusion problem is the sum of polynomials
of degrees n, n — 2, n — 4 ,..., for interior points (see the paragraph following
A9.8)).
Let us consider an ellipsoidal inhomogeneity Q in an isotropic infinite body
(Fig. 23.1). The stress field is calculated when an applied strain (or stress)
before the disturbance due to inhomogeneity is given by
0 ( \ 17 i Г I С _1_ /Т2 1 \
eij\X ) ~~ *^ij ' ^ijkXk ijklXkXl ' V^-'--U
where the coefficients are constants. The eigenstrain in B2.5.1) is assumed to
be
6* (x) = Btj + Bijkxk + BiJklxkXl +¦¦¦.
The strain associated with B3.2) is given by A2.10) and A1.33),
B3.2)
B3.3)
It should be remembered that for points interior to Q, Dijkl is constant, Dijklq
is linear in x, and Dijklqr is the sum of the constants and quadratic functions
Fig 23 1 An ellipsoidal inhomogeneity, indicating points A, B, and С

23. Numerical calculations
189
of x. Using expression A2.16), substituting B3.1), B3.2) and B3.3) into
B2.5.1), and comparing coefficients in the power series, we obtain
(Cstmn - C*mn){DmniJ@)Bij + Dmmjkl@)Bukl+ ¦¦¦}- CstmnBmn
= (C*mn-Cstmn)Emn,
(Cstmn L-stmn)Vmnijk,pV}Hijk+ 4tmnDmnp
= (г* — С )E
V stmn stmn' mnp>
(l/2\)(Cslmn-С*тп)Втпик!р1)@)В^к1+ ••¦ -CstmnBmnpq
= (г* — С )Е
V stmn stmn/ mnpqi
etc.,
B3.4)
Fig 23 2 Ellipsoidal void under a°3 =1, the stress component a[3 at point A in Fig 23 1

190
Chap. 4 Ellipsoidal inhomogeneities
О. O.I 0.2 OS
Р,'а2/а,
Fig 23 3. Ellipsoidal void under a^ = 1, o3^ at point В
3.
2.
-2.
-3.
О. 0.2 0.4 0.6 OB I.
Fig 23 4 Ellipsoidal inhomogeneity under a°} — 1, a[3 at point A, where /i* is the shear modulus
of the inhomogeneity

23. Numerical calculations
191
2.5
/.5
as
-0.5
H=-Q2S
H=O.
n=os
H=l.
н=з.
H=IO.
0.2 0.4 0.6 OS
в -а2/а1
Fig. 23.5. Ellipsoidal inhomogeneity under o°3 = 1, a^ at point B.
where C*w and C,jA:/ are the elastic moduli of the inhomogeneity and the
matrix, respectively. The system of equations B3.4) determines the unknown
В constants. The stress and strain fields are determined from B3.3) and these
constants.
Numerical calculations are performed by Moschovidis A975). Figures 23.2
and 23.3 show a[3 = a303 + a33 at A and В in Fig. 23.1 when a303 = 1 is applied
to an infinite isotropic body containing an ellipsoidal void. The curves in these
figures show that the normal stress a3^ at point В is always larger than that at
point A when аг> а2.
Figures 23.4 and 23.5 show aj3 — a303 + a33 at A and В when a303 = 1 is
applied to an infinite isotropic body containing a spheroidal inhomogeneity
with shear modulus /x*. These curves show the dependency of the normal
stress on shear modulus and geometry. The shear modulus of the matrix is ju,
and Poisson's ratios v and v* are equal to 0.3. For large values of ft*, a[3
shows large compression at point ^4. It is also found that o[x and a[2 are
everywhere much smaller than aj3 except at point A where a? is comparable
to a3r3.
Figure 23.6 shows a2^ = a203 + a23 along the
when a203 is applied
linearly (as shown by the dotted line) to an infinite isotropic body containing a

192
Chap. 4 Ellipsoidal inhomogeneities
Inhomogeneity /
//
/ /
// -'''
//^у=ом
/У*.
/y=»
,,v,
-applied
L
stress
I.
' 0. 0.5 1. 1.5 2. 25
Fig 23 6 Spherical inhomogeneity under linear torsion about the xTaxis
spherical inhomogeneity. The axis of torsion is the x3-axis, and у = /t*//x. It is
important to note that the system of algebraic equations B3.4) to be solved for
5, k, becomes singular when the inhomogeneity reduces to a void G = 0). It is
easy to see this from equation B2.5.1) which, for у = 0, yields
€kl ekl)
B3.5)
As a solution of the inclusion problem, atJ= Cijkl((.kl- (.%,) always satisfies
the equations of equilibrium. Therefore, the right-hand side in B3.5) must also
satisfy the equations of equilibrium. In other words, the system of algebraic
equations has a non-trivial solution only when terms in the right-hand side of
the equations are proper. This situation leads to the conclusion that the system
of algebraic equations B3.4) must be singular for у = 0. This singularity will
be discussed in Section 24.
Two ellipsoidal inhomogeneities
Let us consider two inhomogeneities, &2X and пг, as shown in Fig. 23.7, under
the applied stress 0°. Applying the equivalent inclusion method in domains J2X
and Q2, the equivalency equations become, from B2.5.1),
+ ««) = C,,w(e°, + cw- /?«) in 02,
B3.6)

23. Numerical calculations
193
Fig 23 7 Two ellipsoidal inhomogeneities under applied stress a/}
where C}jkl and filkl represent, respectively, the elastic moduli and the
equivalent eigenstrains defined in Qv The corresponding quantities in Q2 have
the superscript II. It should be emphasized here that ekl is no longer uniform
even if ekl is uniform, since interior points of ii1 are exterior points of п2 and
the stress in Дх is disturbed by Q2.
Let us assume that the material is isotropic and infinitely extended. The
applied strain before disturbance is given by
c°.(x) = EtJ + Eijkxk + Eijklxkx,
B3.7)
where the x{ coordinate system is taken at the center of Qv If the x(
coordinate system is taken at the center of J22, the corresponding expression
becomes
= EU
Eijklxkxl +
B3.8)
Hereafter, tensor components referring to the л:,-coordinate system are de-
denoted by bars.
The two coordinate systems are related by
x, ~ c, =
x, =
B3.9)
where atJ is the direction cosine between the x,-axis and the jcy-axis, and c, is
the л: .-coordinate of the origin б in J22.

194 Chap. 4 Ellipsoidal inhomogeneities
We assume that the equivalent eigenstrains are
fio (*) = Щ + B}jkxk + B}Jklxkx, + ¦¦¦,
Щ{х) = В* + Bfjkxk + В}]к1хкх
, + ¦¦¦.
From A1.33) and A2.10) we can write
Ы*) = D}jki(x)Bki + D}jklq(x)Blkl4 + D}Jklqr{x)B\lqr +
B3.11)
where «Jy and c" are the respective strains caused by/?/- and Щ*. The strain e,
in B3.6) is the sum of e)j and t?.. Interior points of iix are exterior points of
122> and functions D in B3.11) depend on position x as has been discussed
following A2.15). As seen in A2.16), Di(jc) are polynomials of x in flx;
however, Du(x) are not polynomials of x in fij. For д: in fi1; Dn(jc) are
approximated by a Taylor expansion around point 0. Similarly, D1(x) are also
expanded in a Taylor series of л: in &22- Dn(x) are polynomials of x in fl2,
according to A2.16). After these considerations, the system of equations B3.6)
is solved for B. The coefficients of the power series in the left- and right-hand
sides in B3.6) are equated. Consequently, in Qv
stmn mn *
3
Щ
= —AC1 F
^^stmn^mnp

23. Numerical calculations
195
— AC1
дх
= -ACs)mnEmnpq,
etc.
Similarly, in J22,
+
= — ЛГ11 F
stmn mnp'
+
B3.12)
+
]}- cstmnB%n
B3.13)

196 Chap 4 Ellipsoidal inhomogeneities
Э2
ПП \(\\ dII i I r nil
p q
— — AC11 Ё
stmn mnpq>
etc.,
where
д/^i __ /-< s-<\ ЛС^ = С С^ A^ 14Л
Equations B3.12) and B3.13) determine the unknowns B1 and Bn. The
number of unknowns is the same as the number of equations.
Numerical calculations are performed by Moschovidis A975). Table 23.1
shows the stresses at А, С and С (see Fig. 23.8) for the two spherical cavities
in the uniform tension field а°г = cr^ = o303 = 1. The first row of the table
shows the exact solution obtained by Sternberg and Sadowsky A952). The
second and third rows of the table show the current approximations calculated
from the equivalent inclusion method when the equivalent eigenstrains are
assumed to be polynomials of degree zero and of degree one in the position
coordinates. Л is the distance between 0 and 0.
It is found that the interaction between the two cavities becomes negligible
when Л > Aav
Figure 23.9 shows the stress distributions o^ = afj + a. for two spheroidal
cavities (аг = a2 = al = a2 = 1 and аъ = a3 = 0.125) under a303 = 1. The dis-
distance between the cavities is relatively small (Л = 2.5ax), and the interaction
effect is apparent. The solutions have been approximated by using linear
eigenstrains, and the cavities have been regarded as weak inhomogeneities
Ai(I = JnII=10-7Ju.)-
Table 23 1 Two spherical cavities of unit radii in uniform applied tension aj\ = 022 = 033 = 1 at
00, stresses at equators and poles v = 0 25, Л = 4
Point A Point С Point С
oil
Sternberg and
Sadowsky A952) 0.0 150 147 1.51 151 0 0 157 157 0 0
Zeroth degree
polynomial 0.011 1508 1467 1519 1519 001 1534 1534 -0017
1st degree
polynomial 0 001 15079 14676 1503 1503 -0 0008 15495 15495 -0 006

23. Numerical calculations
197
x, с
Fig. 23.8. Two ellipsoidal inhomogeneities under a,°, indicating points А, В and C.
10.,
Cavity I
v -0.3
Д= 2.5
V^gg
I. T5i. 3. 15
x, -axis
-1.5 -1. ^^OJ
\
a
Cavity 1
05/
/
<
V -
Д-
T
аз
2.5
^—
i.
it
i
i
i
f
4
I
i
i
i
i
Fig. 23.9. Two spheroidal cavities (ax = ax = a2 = a2 = 1, a3 = a3 = 0.125) under a°3 = 1.

198
Chap. 4 Ellipsoidal inhomogeneities
2.0
.....I —. ^=====^:=:==\
——~-~^z~—"—~^~--^—
v-v
Д=4
У-га
У-1.5
y.ol'10
Inhomogeneity
1 = i/a = 0.3
0
——-———
30 -20 „„-o«ii -|0 ° ~ l0 20
Fig. 23.10. Two spheroidal inhomogeneities (аг = ax = a2 = a2=\, аъ = аъ = 0.5) under uniaxial
tension Оз'з =1 at oo, o[3 along x3-axis for various values of у = /x'/ji = д"/ц.
Figure 23.10 shows a[3 for two identical spheroidal inhomogeneities (al =
a2 = 1, аъ = 0.5, у = /л1//* = ци/ц, г = j-1 = j-11 = 0.3) under an applied stress
<т3°з = 1. It can be seen that the stress field inside the inhomogeneities is
practically constant.
The accurate elastic interaction between two finite gas bubbles, each of
arbitrary size under excess gas pressure, has been investigated by Willis and
Bullough A969). They found that two such bubbles always attract one another
in an elastically isotropic medium. The Taylor-series method has been recently
used by Johnson, Earmme, and Lee A980) to study cuboidal precipitates.
* 24. Impotent eigenstrains
As mentioned in the paragraph preceding B3.5), the system of equations
B2.5.1) or B3.4) becomes singular when the inhomogeneity is a void, and the
applied stress, before the disturbance due to the inhomogeneity, is in the form
of a degree-one polynomial. The solution of this system of equivalency
equations is not unique. In other words, there are an infinite number of
equivalent eigenstrains which do not generate any stress field within the
matenal. Such eigenstrains are called impotent eigenstrains.
In this section analytical expressions for solutions of the system of homoge-
homogeneous equivalency equations are presented for a spherical void in an isotropic
material. The result of this special case is extended to the general case of
arbitrarily shaped inclusions in anisotropic materials. Most parts of this
section follow the work of Furuhashi and Mura A979).
Let п be an inhomogeneity in D having elastic moduli C*jmn, and let Cijmn
be the elastic moduli of the remainder D - ti. Let a° and e^ be the stress and
strain when the inhomogeneity is absent. Then we have a° = Cijmif°mn. Due to

* 24. Impotent eigenstrains 199
the presence of the inhomogeneity, the stress and the strain are changed to
o° + atj and e9. + ctJ, where
4, = *(«,¦,; + «;,<)• B4.1)
The equivalency equation which transforms an inhomogeneity problem into an
equivalent inclusion problem is expressed by B2.5.1) or
г r* — Ac e = n° — c* e° (ол i\
where e*mn is the equivalent eigenstrain in J2 and
ACljmn= Cijmn-C*mn. B4.2.1)
When 12 has an eigenstrain c*q, the displacement um is expressed by F.1),
um(x)=-jGmkJ(x-x')Cklrge*rq(x')dx'. B4.3)
Substitution of B4.3) into B4.2) leads to the following fundamental in-
tegro-differential equation to give the equivalent eigenstrain ?*:
О^кт(х-х')а^(х')йх' = до(х), B4.4)
where
°* = Cljmne*mn, д^ = о°-С*тпе°тп. B4.5)
We can calculate the stress a,, from the formula
ч
B4.6)
Let us consider a special case in which the inhomogeneity Q is a spherical
isotropic inhomogeneity with shear modulus ju,* and Poisson's ratio v* and the
matrix is an infinitely extended isotropic material with shear modulus ju, and
Poisson's ratio v.

200 Chap. 4 Ellipsoidal inhomogeneities
The applied stress field before the inhomogeneity disturbance is assumed to
be a homogeneous polynomial of degree one,
B4.7)
where B{ are given constants.
It is well known that when J2 is an ellipsoid, the integral
B4.7.1)
becomes a polynomial of degree one without the constant term for interior
points, as discussed following A2.16). Therefore, we can write
We seek a solution of the integro-differential equation B4.4) in the forms,
o*{x)=Aijpxp, AiJp = AJip, B4.9)
where Aijp are constants to be determined. Substituting this expression into
B4.5) and using B4.8), we obtain
jqq %px4 = Bijqxq. B4.9.1)
Hence, Aijq must be a solution of the following linear system of equations:
A,jq + ^Ak!p = Bljq. B4.10)
The number of equations in B4.10) is 18. If U is not a void, the independent
number of Bijq is also 18. From B4.7) and B4.5), we have
?<7,, = %=-QWl,,> B4-u)
since afj = 0.
When Q is a void, we get the three relations BtJJ = 0. Therefore, when ?2 is a
void, the maximum number of independent coefficients Bijp is 15. The system
of 18 equations B4.10) consists of four groups of equations with coefficients
(^nii' ^221' ^33i> ^122' Л133), (A222, An2, Аъгг, Al2l, А2ЪЪ), (Аъъъ, Апъ,
A223, Ain, A232) and (A123, A132, A231). Each of the first three groups is made

* 24. Impotent eigenstrains
201
up of a 5 X 5 matrix, and the last group is made up of a 3 X 3 matrix. If Q is
not a void, the solutions of these equations are uniquely determined.
When J2 is a void, the following matrix belongs to the first three groups:
1
35A -?)
8-14? 3-7? 3-7? -8 -8 \
3-21? 24-28? l-lv -8 2-14?
3-21? 1-7? 24-28? 2-14? -8
-4 + 7? -4 + 7? 1 13-7? -5 + 7?
\-4 + 7? 1 -4 + 7? -5 + 7? 13-7?
Similarly, the matrix belonging to the last group is
j /23-21? -5 + 7? -5+7?
— г -5 + 7? 23-21? -5 + 7?
35A-") _5 + 7? -5 + 7? 23-21?
B4.12)
B4.13)
The ranks of these two matrices are 4 and 3, respectively. Hence, we see that
the rank of the matrix of coefficients in equation B4.10) is 15 when Q is a
void. Therefore, the homogeneous equations
belonging to B4.10), have three independent non-trivial solutions Ask/p (s =
1, 2, 3). Hence, if we denote by Aklp an arbitrary particular solution of the
system of equations B4.10), we can express the general solution as
s-l
klp,
B4.15)
where Cs are arbitrary constants. We can easily ascertain from B4.12) that
A\t is the following vector:
221 Л331
B4.16)
Similarly, A\lp and A\lp are obtained by the cyclic permutation of the indices.
The eigenstrain e*1 corresponding to this solution is obtained from B4.5) and
B4.9) as
e22
_ ,*i _
~ €33 —
*i =
?13
,*1 _
€23 ~"
B4.17)

202 Chap. 4 Ellipsoidal inhomogeneities
We obtain t*j and e*f by the cyclic permutation of the indices. It can be
shown that the eigenstrain components in B4.17) do not introduce any stress
field in D.
We conclude, therefore, that the eigenstrain B4.17) is impotent.
If the applied stress is the pure torsion around the *3-axis, i.e. a203 = x1 and
ст1°з = ~xi> ^ is easily verified from B4.13) that the following vector is the
particular solution of equation B4.10):
i132=-f, A231 = l B4.18)
with all other components equal to zero.
We have considered a rather special case by choosing a sphere and an
isotropic material. We can generalize the results for cases where the shape of
the inclusion and the anisotropy of the material are completely arbitrary. The
preceding example of an impotent inclusion suggests the following generaliza-
generalization.
Suppose that eigenstrain components «*. in ft are derived from functions
u* by
«?,=*(«?„+«;,*)• B4-19)
It is assumed that u* are continuous in D and sufficiently smooth in the
interior of ft. Then, if u* = 0 on the boundary | ft | of ft, the stress compo-
components atj vanish everywhere in D and this eigenstrain is impotent (ft is an
impotent inclusion).
Before considering the analytical proof of this theorem, we make an
intuitive observation: Since the boundary displacement is zero, the remainder
D - ft is not disturbed by ft; hence, the displacement and stress in D — ft
vanish. On the other hand, the eigenstrain defined by B4.19) is compatible in
ft.
The analytical proof is as follows: When e* is given by B4.19), the use of
Gauss' theorem leads to
j P)
^JX-X')u*{xl) йх' B4-20)

* 24. Impotent eigenstrains 203
where np(x') stands for the outward normal at the point x' on | fl |. Because
и* = О on \Q\ and because of the definition E.2) of Green's function Gmk,
B4.20) becomes
Hence, B4.6) leads to atJ = 0 everywhere.
When the boundary of Q is expressed by an implicit function f(x) = 0, we
construct a function u* explicitly. Assume that / is sufficiently smooth.
Define the function / as
/(*). *ei2' B4.22)
O, xeD-J2.
Let gi be a sufficiently smooth arbitrary function and set
uf(x)=gi(x)fXx). B4.23)
и*(я:) is continuous in D but, in general, is not differentiable on the boundary
of Q. If ?2 is an ellipsoid defined by equation A1.2), then the function u* has
the form
Х "' B4.24)
where g, is an arbitrary smooth function. Since g, is arbitrary, we can
construct an infinite number of impotent eigenstrains from B4.23) and B4.19).
If (*j is a polynomial eigenstrain of degree one, the function u* correspond-
corresponding to this eigenstrain is a polynomial of degree two and can be expressed as
uf(x) = gt(aJkxjXk + ajXj + a0), B4.25)
where a]k, aj, a0, and g, are constants. The eigenstrain derived from this
function is impotent if, and only if, the domain ?2 is bounded by the quadratic
surface
a xx+ax+a=0 B4 26")

204 Chap. 4 Ellipsoidal inhomogeneities
It is well known that there is no bounded quadratic surface except the
ellipsoidal surface. Hence, if c*j is a polynomial impotent eigenstrain of degree
one in a bounded domain ?2, then ?2 is an ellipsoid.
Similarly, if ?*y is a uniform impotent eigenstrain in a domain ?2, then ?2 is
a half-space.
There are not only smooth impotent eigenstrains, but also discontinuous
ones. Let ?2x,...,?2j be subsets of D. Suppose that u*p is continuous in D,
equal to zero outside ?2p, and sufficiently smooth in the interior of ?2 . Since
the eigenstrains t*p derived from these functions are impotent in ?2p, the
eigenstrain e* = ejy + • • • + e*/ is also impotent in the union of ?2l,...,uJ. If
the interior of intersection ?2r П ?2q for integers г Ф q is not empty, this
eigenstrain is discontinuous on the boundary of ?2rP\ ?2q. For example, if we
define the function uf as
B4.27)
then the impotent eigenstrain derived from this function has J — 1 discontinu-
discontinuous surfaces in the unit sphere \x\ < 1.
25. Energies of inhomogeneities
The elastic strain energy, potential energy, interaction energy, and the Gibbs
free energy will be discussed for a body which contains inhomogeneities.
Elastic strain energy
First we consider the case of a body D containing inhomogeneous inclusions
and free from any external force or surface constraint. A given eigenstrain in
?2 is denoted by ef-, where ?2 is the subdomain of D occupied by the
inhomogeneous inclusions. The (elastic) strain energy is the same as A3.1),
B5.1)

25. Energies of inhomogeneities 205
where etJ = e^- efj and etj¦= \(utJ + u] t). Integrating by parts, we obtain
from B5.1)
jtfj &D. B5.2)
If 0 is an ellipsoid and «fy is uniform, B5.2) becomes
W*=-\Vol}t!j, B5.3)
where V is the volume of fi and atj is given by B2.13.1).
If the elastic moduli C*jkl in U are slightly different from Cijkl of the
matrix, B5.2) is approximated by the quantities defined in a homogeneous
inclusion problem in which Q has the elastic moduli Cijkl and eigenstrain ef¦.
The displacement, elastic strain, stress, and elastic strain energy for the
homogeneous inclusion problem are denoted by u\, e'tJ, а'и and W, respec-
respectively. Then, A3.3) is written as
dD>
where ui} is artificially subtracted and added, and the integral domain п is
extended to D since efj¦ = 0 in D — Q. ut is the displacement in the inhomoge-
neous inclusion problem leading to B5.2). Since tfj - \{ut j + м;,) = -etj and
B5.3.2)
Similarly, B5.2) is rewritten as
B5.3.3)
by the use of relations e?• = \{u'Uj + u'j _,.) - e'tj and JDaiju'i,j dZ) = O. Since

206 Chap. 4 Ellipsoidal inhomogeneities
aIjeij = aueV in D~°>
W*-W = \([ аие'и dD-f a'etJ dD ). B5.3.4)
In ft,
where ACljkl = C*Jkl - Cijkl. Then
W*-W' = \JACijklekle'u AD. B5.3.6)
When ekl = e'kl + Aekl, and ACi]kl and Aekl are small quantities with respect to
Cijkl and ekl, B5.3.6) is approximated to
W* -W' = \[ ACukle'kle'u dD B5.3.7)
which was first proposed by Eshelby A966). Since quantities e'u, o'i} and W
can easily be obtained from the homogeneous problem, B5.3.7) can be used
for an approximation of W* for the inhomogeneous inclusion problem. In the
above equation, ft can be the sum of many inhomogeneous inclusions, i.e.
ft = Qx + Q2 + ....
If ft consists of two ellipsoids, iix and ft2, W* has the same form as B5.2),
ft = ftj + ft2, and a,7 = off + off, where a^m) is the stress caused by efj in ftm.
The interaction energy between ftxand ft2 is — fap^hfj dD and which is
equal to - JapWefj dD.
Next, consider the case when a body D, containing an inhomogeneous
inclusion ft with eigenstrain if., is subjected to an external surface force Fr
The displacement field is denoted by м° + м, and the stress field by a° + aiy
uj and a°j are the displacement and stress when Ft acts in the absence of ft.
Then
^ = n{o?, + oil){ull + ul-e')dD, B5.4)

25. Energies of inhomogeneities 207
where
°?j=CijkluokJ, o' + Ou-Cfjuiulj+Ubj-tZ,). B5.5)
Here Cijkl and C*Jkl are the respective elastic moduli of the matrix and
inhomogeneous inclusion.
The expression for W* given in B5.4) may be further simplified. An
equation similar to A3.6) holds,
{ul + uJdD^O. B5.6)
Furthermore, we have
= оии1 + о°е*р B5.7)
where «*; is the fictitious eigenstrain introduced1 in the equivalent inclusion
method and
°о = Су„(ик>,-??,-??,) in 0. B5.8)
Thus, we have
?(«,,,• - *fj) dD =[ofc dZ>, B5.9)
Si
since jD Ojjufj dD = 0. The elastic strain energy, therefore, becomes
W* = \ f o°u°j dD + U „fa dD-Пo^ dD. B5.10)
If Q, is an inhomogeneity (efy = 0), the elastic strain energy is
B5-11)

208 Chap. 4 Ellipsoidal inhomogeneities
On the other hand, when the applied stress or?, is absent and i2 is an
inhomogeneous inclusion with eigenstrain efj, the elastic strain energy be-
becomes
W*=-\l oueP dD. B5.11.1)
Ja J
Therefore, the elastic strain energy B5.10) for the case when a? and efj coexist
is the sum of B5.11) and B5.11.1) because of Colonnetti's theorem A921) or
B5.28); namely
W* = \{ of>ulj dD + \ [ofcj dD - hjo^fj dD B5.11.2)
D 02 Q
The above expression has the same form as B5.10), but e? and atJ in the two
expressions are defined differently. €*, in B5.11.2) is the solution of B2.7), and
atJ in B5.11.2) is the stress obtained from B2.13.1).
Interaction energy
When a body D contains an inhomogeneity Q and is subjected to Ft on S, the
total potential energy of the body (the Gibbs free energy) is defined by the
sum of the elastic strain energy and the potential energy of Fp
= * jTa°«°, dD + ф°е* dD - JFt{uf + ut) dS, B5.12)
from B5.11).
If the material is homogeneous, W becomes
ul dD - JFrf dS. B5.13)
The interaction energy between Ft and the inhomogeneity is defined by
~W-W0=\ fa?,e* dD - />,u, dS. B5.14)
JQ ' JS

25. Energies of inhomogeneities 209
It holds that
JFiUi dS = fo°.njUl dS=f^JuiJ dD
and
f °?jKj -«J)dfl = / С^и^Ди,,, - c* ) dD = / „?,„„ d?> = 0. B5.16)
J D J D JD
Therefore, we have
°<*jdD. B5.17)
If a° is uniform and J2 is an ellipsoidal inhomogeneity, e*7 also becomes
uniform and
AW=-\Vo°t*1, B5.18)
where V is the volume of Q.
Let us consider 4fF when i2 is a crack. As shown in Chapter 5, B5.18) is
used for the derivation of Griffith's fracture criterion, where V e* is kept
constant when V-*0 and «*• -» oo. For later discussions on the crack opening
displacement and crack growth rate, it is useful to derive an alternative
expression for AW here.
When B5.13) is subtracted from the first expression in B5.12),
"i,j dD - fs°>u«i dS- B5.18.1)
In D, a°Juij = OjjU^j since domain U has no volume. Q has only surface 2
where ut is discontinuous. It holds that
jatJulj dD = foljnju4 dS - joiJjU° dD = 0 B5.18.2)

210 Chap. 4 Ellipsoidal inhomogeneities
because uf is continuous on 2, and
/а°И,и, dS = / afJjUi dD + f o°utJ dD - /а°п,[и,] dS
»,]dS, B5.18.3)
where rij is the outward normal to the upper surface of the crack (я = @, 0, — 1)
when the crack is on the x^-plane) and [ut] is the crack opening displace-
displacement, i.e.
Aut = [и,.] = иДиррег) - M.(lower). B5.18.4)
We have also
о- и ¦ ¦dZ)== i o-n и ¦ dS ~\~ j o- n \u I dS — | ex- и ¦ dD
, ч ''' Js lJ J Js 1J J ' JD IJ-J
= joiJnJ[ui] dS= - f a^rijiu,] dS B5.18.5)
since (a°. + otj) rij = 0 on 2. Finally, therefore, we have
AW=\fa°nJ[ui]dS B5.18.6)
where 2 is the crack surface.
When a body D contains an inhomogeneous inclusion п with eigenstrain
efj, and it is subjected to Fjf the Gibbs free energy of the body is
:(M,° + M,)dS. B5.19)
This total potential energy is compared with Wo as defined by B5.13). The
difference between W and Wo is
AW=W-W0. B5.20)
Since the first term on the right-hand side in B5.19) can be expressed by
B5.10), we have
AW= \ f a°ef, dD - \ f auef dD - ('Ftut dS. B5.21)
Ja Ju Js

25. Energies of inhomogeneities 211
It also holds that
JFiUi dS = fofcu, dS = jo?jUiJ dD
= fo°(uiyJ-tfj-tfj)dD-
and
/ 0. B5.23)
JD
Therefore, we have
AW= - \ \ o°e* dD - H aut\: dD - / af.tf. dD. B5.24)
Ju JU JQ
Colonnetti's theorem
The variation of the Gibbs free energy associated with a change in a given
physical system plays an important role in certain cases, as shown in Chapter
7.
Suppose that a body D containing inhomogeneities Q is subjected to an
applied force Ft on the boundary S of D. The Gibbs free energy of the body is
expressed by B5.12) or
W=W*+V*, B5.25)
with
B5.26)
where afj = a° + atj, e/J = \(u?j + ufyi)> and uf = uf + ut. efj is is the elastic
strain, W* the elastic strain energy, and V* is the potential energy of Fr If the

212 Chap. 4 Ellipsoidal inhomogeneities
inhomogeneities in D are further subjected to the eigenstrain efj, the Gibbs
free energy of the body changes to B5.19). The new stress, elastic strain, and
displacement become a* + aj, efj + efj, and uf + uf, where a", efj, and uf
are increments (changes) in the fields due to efj. The Gibbs free energy of the
body B5.19) can now be written as
W=W*+V* B5.27)
with
B5.28)
V* = - ( Ftuf dS- (Ftuf dS.
Js Js
This result has been obtained from Colonnetti's theorem A921) as shown
below. The elastic strain energy is
= hjofctj dD + i j\,,% dD + jo»e?j dZ>, B5.29)
since о*е* = Cfcefa* = oBkleAkl, where C**kl is CiJkl in D - Q and C*kl in 0.
On the other hand,
/ о*е* dD = J o»u?; dD = fo'njuf dS - f a^uf dD = », B5.30)
JD JD JS J D
since o?nj = 0 on S and o?j = 0 in D. Equation B5.30) is Colonetti's
theorem. The variations of W* and V* associated with the change from the
condition corresponding to B5.26) to the condition corresponding to B5.28),
are
AW* = \jofe* dD = hjo*{u*j - eg) dD = -\ja^fj dD, B5.31)
AV* = - (Ftuf dS = - [ofiufn, dS= - f ofiufj dD
Js Js JD
Aefj + efj) dD-- jofcfj dD B5.32)

25. Energies of inhomogeneities 213
or
AV* = - [о°и*п, dS = - / a,X, dD
where e** has been defined in B2.15). Since
{<j ' «?/) dD = //L< dI) = 0, B5.34)
B5.33) leads to
4K*= -fo°j€** dD. B5.35)
f
The terms \\оа^е^ dD-jsFtuf dS have been calculated from B5.12).
Thus, we have
AW=W-W0= -\f°?jt*tj *D~ ^f°H d# - fjyfcfj dD, B5.35.1)
where Wo is defined by B5.13) and c*^ is e* in B5.17) or B2.5), which is the
equivalent eigenstrain for the inhomogeneity under applied stress o,°. The
above expression B5.35.1) is an alternative expression of B5.24).
The expressions of the interaction energies derived so far are valid also for
the interaction between an inhomogeneous inclusion Q and a different internal
stress, afj and uf in the previous expressions are interpreted as the internal
stress and displacement caused by the other internal stress source.
Uniform plastic deformation in a matrix
As an application of Colonnetti's theorem, let us consider a composite
material under an applied force Fr The material D consists of inhomogeneities
О and matrix D — Q. The elastic stress is denoted by a/*. Assume that the
inhomogeneities are stronger than the matrix and that a uniform plastic strain
tfj is distributed in the matrix. Due to the plastic strain misfit with the

214 Chap 4 Ellipsoidal inhomogeneities
inhomogeneities, the stress state changes from o* to o* + aj. Then, the Gibbs
free energy of the system becomes
W=W*+V* B5.36)
with
B5.37)
V* = - (Ftuf dS- (Ftuf dS,
where efj and ef} are the elastic strains associated with afj and af}, and uf
and uf are the corresponding displacements. For the discussion in Section 43
we will investigate the dependency of W on e?.. The first terms in B5.37) are
independent of €^. The second terms can be rewritten as follows:
dD = *
- (Ftuf dS= - /a,X«,dS= - / o^ + ef^dD B5.38)
ij ij
Since cf7 = 0 in i2 and efj is uniform in D — Q, the integrals in B5.38) are
actually defined in D — U. However, it is convenient to extend the integral
domain into D and subtract J2 from D. Then, since fD of} dD = 0 (see D2.3)),
we can write
B5.39)

25. Energies of inhomogeneities 215
Since a* = a°. + atj and fD atJ dD = O,
(№dD-
- -'fj«°jfD iD + ,ф,А, dD. B5.40)
Thus, the part of W which depends on efj becomes
Л W = iefy/ a* dD- ^о°( dD + ef.j a* dD. B5.41)
a;8 can be calculated as the eigenstress by assuming that i2 has eigenstrain
— ef and that the matrix has no plastic strain, since the misfit strain is a
relative quantity between the matrix and the inclusions.
Energy balance
In order to understand the physical meaning of the Gibbs free energy change
AW, we consider the energy dissipated when efj in B5.27) or B5.36) is
changed by Sefj. The work done by applied force Ft during the change is
f^Suf dS, B5.42)
Js
where Ft is kept constant during the variation. The change of elastic strain
energy is
B5.43)
since efj does not change during the variation, o* is the stress due to a/J and
the inhomogeneity, and a? is the stress caused by efj and the inhomogeneity.
The energy SQ dissipated during the variation is the difference between
B5.42) and B5.43),
8Q = JFfiuf US- j (о* + о»)8е* dD, B5.44)

216 Chap. 4 Ellipsoidal inhomogeneities
where
*(«!!*,+ ««*,.)= ве*+ «??,.. B5.45)
Thus, we have
SQ=-S(AW) B5.46)
which means that the change in the Gibbs free energy is minus the energy
dissipated during the variation 8efr
Expression B5.44) can be rewritten as
8Q= [(ofj+ofiScfjdD, B5.47)
in which B5.45), (o* + о?)п^ = Ft on S, and olj = 0 and a/jj = 0 in D are
used. Expression B5.47) can also be obtained from B5.46) and B5.35.1), since
a? in B5.35.1) is a linear function of efj as seen from B2.13.1).
When efj is defined uniformly only in the matrix, B5.47) becomes
[ (
D-Q
B5.48)
From D2.3) and the definitions of afj and a?, we have
f o* dD = 0, ( of* dD= f <r° dD. B5.49)
jd jd jd
Therefore,
SQ = SefJfo* dD - f(o* + a*) d? ), B5.50)
which can also be obtained from B5.46) and B5.41). a* + aj = a/J. + atj can
be calculated from B2.13), where efj is taken as — cfj in Q, since atj is
equivalent to the eigenstress (stress disturbance) caused by the misfit — ef in
?2 and the applied stress a°.

25. Energies of inhomogeneities 217
Let us consider an example of the simple tension a303. It is assumed that the
matrix is a perfectly plastic solid with the yield stress ay and that the
inhomogeneities u = lLpup are perfectly elastic. The plastic strain has no
dilatation and t[x = e%2 = ~€зз/2- The energy dissipated during Sefj is
SQ= f oS€P3dD. B5.51)
JD-a
Then, B5.50) becomes
o°J dD - f (a33 - \an - \o22) Ad). B5.52)
The last terms are calculated from B2.13), resulting in
a33 - Han + a22) =Лс& + 5a°3, B5.53)
where
A=E*/2(l-v*), B={E*v-Ev*)/E{\-v*), B5.54)
for penny-shaped particles (the disk plane is perpendicular to the x3-axis);
A = \E*E{1 -5v)
E*(l + v)(8 - 10»-) + EC? - 5»<)A + v*) '
C25 551
E*{\ + v){
for spherical particles; and
|?*{3?A - 2v*) + 2E*(l
~ E*(l )
v*)(l-2v*) '
B5.56)
v)/E - 2E*v*{l + v)- E(l + v*)(l - 2v*)
for rod-shaped particles (the axis of the rod is parallel to the x3-axis). The
values of A and В for anisotropic media have been calculated by Lin et al.
A973). When B5.53) is substituted into B5.52), we have
-f)ay = A -/K3 -f(At& + Ba3°3) B5.57)

218 Chap. 4 Ellipsoidal inhomogeneities
or
°э°з =
1-/A+5) 1-/A + 5)
B5.58)
where / is the volume fraction of the particles. Equation B5.58) has been
obtained by Tanaka and Mori A970). The stress-strain relation of the com-
composite material displays linear work-hardening as shown in Fig. 43.1. The
details are presented in Section 43. The yield point also increases since
A -/)/{! ~ A + B)f) > 1- It decreases, however, when E* < E and v* ~ v.
Let us consider a creep problem under a constant applied stress a303. If we
assume that the matrix behaves as a Newtonian viscous solid (the stress is Д
), SQ in B5.50) can be written as
B5-59)
then B5.50) becomes
A -f)iiif3 = A -/)o3°3 -f(A<b + Bo°3). B5.60)
The above differential equation has the solution
B5-61)
when the initial condition ?^3 = 0 at / = 0 is used. The composite material, as a
whole, behaves like Kelvin's material (Mura 1972). Cyclic creep problems have
been investigated by Mura, Novakovic, and Meshii A975) and Shetty, Mura,
and Meshii A975) by using equation B5.50).
26. Precipitates and martensites
Precipitates and martensites in alloys are typical examples of inhomogeneous
inclusions. The elastic strain energy caused by these particular inclusions will
be discussed in this section. If interfacial energy can be neglected, the
morphology of precipitates and martensites can be determined by the condi-
condition that the elastic strain energy B5.2) take a minimum value. otJ in B5.2) or
B5.3) can be evaluated by the equivalent inclusion method described by
B2.13), where ?°y = 0 and a° = 0, since no applied load is considered here.

26. Precipitates and martensites 219
Isotropic precipitates
Consider an infinite isotropic matrix with shear modulus ju. and Poisson's ratio
v, containing an ellipsoidal precipitate whose elastic constants are ц* and v*.
Barnett et al. A974) calculated the elastic strain energy per unit volume of the
precipitate when
e'j~ "*' , B6.1)
ax=a2 = a, 0 < /? = аъ/а < oo,
where € is a constant misfit strain. Their results are
1 — v
where
у = цA + v)/{\ - 2v), у* = ц*A + v*)/{\ - 2v*),
M = 1/A - p2) - /3A - jS2)/2 cos/?, Д < 1
= 1/3, /8 = 1 B6.3)
= 1/A - Д2) + /J@2 - 1)"V2 cosh^ /8 > 1,
if = A + (м*/М - 1){ME - 4^) - 3CM- 1)/2A - 02)}/2(l - v))
X (l + (fi*/^ - 1){A - 2v) + 2MA + v)
When ju* = ц and v* = j», B6.2) reduces to A3.21),
v). B6.4)
The normalized strain energy W*/Wo is plotted in Fig. 26.1 against /} for
several values of ju*/jh. Barnett et al. A974) concluded that the minimum

220
Chap. 4 Ellipsoidal inhomogeneities
2.0
1.5
10
0 5
n
I ii,
" \ A* = 1 00 = A
\ v' = 0 308=t/
\ /x = 7 54 x 10" dynes/cm2
\ 0 - o3/q, a, = q, = о
N.
^^^^ /x* = 3/j.
1
—
0.5
1.0
Fig 26 1 Normalized strain energy of an isotropic coherent ellipsoidal precipitate in an isotropic
matrix vs aspect ratios A, v, and p> are the anisotropy ratio, Poisson's ratio, and shear modulus,
respectively
strain energy is achieved by a flat ellipsoid when ju,* < ju, and by a sphere when
ju* > ju. The same conclusion has been stated by Laszlo A950).
Anisotropic precipitates
Here we calculate the elastic strain energy when an infinite anisotropic matrix
contains an ellipsoidal anisotropic precipitate. The precipitate is an inhomoge-
neous inclusion with eigenstrain «?.. The dependency of the results on the
orientation and shape of the precipitates is discussed.
Let us denote the elastic moduli of the matrix by Cijkl and of the
precipitate by Cfjkl. The jc, coordinate system is taken along the crystallo-
graphic directions of the matrix ([100], [010], [001]). The orientation of the
precipitate is characterized by в and ф (see Fig. 26.2).
Mori et al. A978) have obtained contour lines of the elastic strain energy
for a needle-shaped precipitate (Fig. 26.3) and those for a disk-shaped pre-

26. Precipitates and martensites
221
Fig 26 2 Orientations of precipitates with respect to crystallographic directions x,
cipitate (Fig. 26.4), when the matrix is copper (Cu = 16.84 X 1010 N/m2,
C12 = 12.14 X 1010 N/m2, C44 = 7.54 X 1010 N/m2) and
?S = 8./' C*=/Co- B6-5)
These conditions assume that the phase transformation strain is dilatational
Needle (icCj/m1)
(a) f = 0.3
(b) f = 1.0
(c) f = 3.0
9/45 9.6 Ю.О Ю.4 Ю.69 16.09 17.0 19.0 2048 24.9 26.0 30.0 34O 365
br
Fig. 26.3. Contour lines of energy; the matrix is copper and C* = /C,7.

222 Chap. 4 Ellipsoidal inhomogeneities
Disk ixfj/rrf)
(o) «'0.3 (b) fl.O (C) t.3.0
3.44 4.0 50 60 6SB П.5 140 ».O 22O 233 34Л 400 SQO SQO (№9
70.0
Fig. 26.4. Contour lines of energy; the matrix is copper and C* = fCtj.
and the precipitate crystalline directions coincide with those of the matrix.
It can be seen that the minimum energy is achieved when the disk plane is
perpendicular to [001]. This is due to the fact that the Zener anisotropy factor
(Zener 1948)
A = 2C44/(Cn-C12) B6.6)
is greater than 1 and C^ is assumed to equal fCtj in the energy calculation.
The stress components in an ellipsoidal inhomogeneity containing eigenstrain
efj are expressed by B2.13.1). The right-hand side in B2.13.1) becomes A7.47)
for a flat ellipsoidal inclusion, where «** is used for ?*y. As explained by
A7.37.1) and A7.42), the displacement and the traction at the flat interface are
zero. Using the principle of equivalent inclusion, we can find that the
displacement and the traction at the flat interface of the inhomogeneous
inclusion are also zero. This means that the inclusion is stressed but the matrix
is not. The stress state is independent of the matrix moduli and remains the
same even if the moduli of the matrix are equal to C*jkl. Therefore, this
inhomogeneous inclusion becomes the homogeneous inclusion with efj. The
stress is
°
РЧ
B6.7)
from A7.47), where Nu(?) and Z>(?) are expressed in terms of C*kh and (¦ is
the unit vector normal to the flat surface. The fact that the elastic stress state
is independent of the matrix elastic moduli has been established by Mura and
S.C. Lin A974) and Lee, Barnett, and Aaronson A977) who have pointed out
that the elastic strain energy of a disk-shaped precipitate is independent of the

26 Precipitates and martensites 223
elastic moduli of the matrix. When C*kl has cubic symmetry and efj = S^e*,
the elastic strain energy W* is calculated from B6.7) and B5.3),
W* = \V{C*l + 2C*2)(e*J{3 - (C* + 2C*2)F}, B6.8)
where
F = {C?2 + 2CU(Cb - C*2 - 2С*,)Г + 3@^ - C*2 - 2C&J
X {C*Q2 + C&{C*X + Cf2)(C* - C*2 - 2Q)r
+ (Q - Cf2 - 2C*,J(C* + 2C*2 +СЬ)Д}~\ B6.8.1)
It can be seen that B6.8) has an extreme value when either one of \ix\,
| ?2|, or 1131 is equal to 1 or | ?~ | = 1121 = | ?, | = 1//J. When the Zener
factor A* = 2Q/(C* - C*2) > 1, the minimum of W* is found at ? = @, 0, 1),
which is
W* = К(с*J(С* + 2C*2)(C* - C*2)/C*i; B6.8.2)
this is in agreement with the result of Mura and S.C. Lin A974). When
A* < 1, W* takes a minimum at ? = A/Д, 1/yfr, l/i/З").
From Fig. 26.3, it is found that the elastic strain energy of a needle-shaped
precipitate is a minimum when the needle is aligned parallel to one of the
crystallographic directions @01). However, the minimum of the elastic strain
energy of the needle can not be determined by A and A* alone. In fact, Mori
et al. A978) have shown that the orientation of the needle (the axes direction)
for minimum energy (for A > 1, A* < 1) is [111] or [001], depending on the
combination of Ctj and C*.
It is important to note from B6.7) that conclusions A7.42) and A7.46) with
condition A7.44) are obtained even for an inhomogeneous, flat ellipsoidal
(disk-shaped) inclusion, where e*y is replaced by efr The strain energy calcula-
calculation of the spinodal alloy is essentially identical to that of a disk-shaped
precipitate. The elastic strain energy of the spinodally decomposed cubic alloy
has the same orientation dependence as B6.8) (see e.g. Hilliard 1970). For

224 Chap. 4 Ellipsoidal inhomogeneities
example, when a solute concentration, c, fluctuates along the direction ?, с
and the corresponding eigenstrain are expressed by
00
c= ? c(n) cosBvnx/X),
c(n) cos
B6.8.3)
where X is the wave length of the fluctuation and jc is the distance along ?.
Applying C.14) and B5.2) with «^ = c*6iy, we obtain the elastic strain energy,
W* = K^XQ + 2C*2)(e*J{3 - (C*x + 2C*2)F} B6.8.4)
per unit volume of a cubic crystal. Here, (c2) is defined as
<c2) = (l/X)fXc2 dx B6.8.5)
and F has been defined in B6.8.1). It has often been said that the coefficient
G* = (C*x + 2C*2){3 - (C*! + 2C*2)F] in equation B6.8.4) is the elastic mod-
modulus in the direction along which the concentration of solute atoms fluctuates.
However, such a concept leads to a misunderstanding, since the stress compo-
components normal to the disk surface (a3p) are zero as is shown by A7.42), while
G* is simply evaluated from B6.8). This elastic strain energy is caused by the
stress components parallel to the disk surface (оц, о22, o12). This energy is
somewhat related to the elastic modulus perpendicular to the direction of
solute atom fluctuation.
One can extend B6.8.3) to the situation where e°y is not a pure dilatation.
Such a case might occur when a carbon atom concentration fluctuates in
martensitic steel that has a high carbon concentration. In this case, because of
the interaction of the carbon atoms, almost all the carbon atoms are accom-
accommodated in a particular type of sublattice (Zener 1946, Mori and Mura 1976,
Mori, Cheng, and Mura 1976). For example, e?. in B6.8.3) has components of
?оз ф ^ = ?o2 It is known that ?°3 = 0.93 and e^ = c°22 = -0.08. If the con-
concentration of the carbon atoms fluctuates as expressed by B6.8.3), we obtain
the elastic energy per unit volume,
W* = \(c2){H-G), B6.8.6)

26. Precipitates and martensites 225
from C.14) and B5.2), where
Я = {2Cu(a° J - 4С12а°Л°э + (Qi + C12)(a°J}
:n + 2C12)(Cn-C12)}~\ B6.8.7)
2(CU - C^)^ - C12 - 2^)
2^X3(^2 + C^X C^ + H)ij + 2(CU - C12 -
- c12 -
x {(c^J^! + ^(^ - c12 -
12
with
° =
—
(cu - c12 - гс^)^^ + 2C1 j^lil} ~\
B6.8.8)
It can be shown that W* has maxima at ?2 = 1, ?2 = 1 or ?2 = 1 and minima
at
?~2 = sin20, ?-2 = 0 and |2 = cos2^ B6.8.9)
or
l? = 0, || = sin2^ and |2 = cos26 B6.8.10)
with 0 given by
B6.8.11)

226 Chap. 4 Ellipsoidal inhomogeneities
where A is the root of
12
(Cu - C12 - 2C44)(C11 + ^^
-2[C44C11{(C11 - C44)(a1°1J + (Cn - C44)(a303J - 2(C1
-(Cu - C12 - 2C44)(C11 + C^QK J]л + ^^{(a»J + (a°2J}
- ^^{(q, - C44)(a1°1J + (Cu - C44)(a3°3J - 2(C12
+ (Cn ~ C12 ~ 2C44)(C11 + C12)C^(cnJ = 0 B6.8.12)
{0<A <1).
By putting Cn = 2.37 X 10й N/m2, C12 = 1.41 X 1011 N/m2 and Q, = 1.16 X
1011 N/m2, one obtains в = 19.3°. This example has been treated by Toyoshima
A980) for a discussion of the modulated structure observed in iron-carbon
martensites (Toyoshima and Nagakura, 1979). Toyoshima has found that the
direction of the fluctuation of the carbon atom concentration is expressed by
either [hoi] or [ohl], and the angle в depends on the amount of carbon in the
martensite; as the carbon concentration goes to zero, в approaches 22°.
Incoherent precipitates
The precipitates discussed in the last two subsections are coherent precipitates.
When a second phase is nucleated, the interface between the precipitate and
the matrix is classified as either coherent or incoherent. By coherent, it is
meant that there is a one-to-one correspondence between atoms at the
interface; an incoherent interface refers to a lack of conservation of lattice
sites across the interface. For an incoherent precipitate, the stress is purely
hydrostatic along the interface. The associated elastic strain energy has been
obtained by Nabarro A940) for the isotropic case and by Kroner A954) and
Lee and Johnson A978) for the anisotropic case.
Let us consider an ellipsoidal incoherent precipitate in an infinite aniso-
anisotropic medium. The elastic moduli of the precipitate and the matrix are

26. Precipitates and martensites 227
denoted by C*k/ and CiJkl, respectively. The equivalent inclusion method is
employed. From B2.11) we have
C?jkl(tkl-tl,) = Cijkl(ekl-€ll-ctl), B6.9)
where no applied stress is considered. The stress components given by B6.9)
must be hydrostatic,
Furthermore, the incoherent precipitate is characterized by
<& = «• B6.11)
This means that for the incoherent precipitate the six components of efj are
not given, but only <.pkk is prescribed to be a constant, €. By assuming the
relation B2.12), the equivalent eigenstrain t* can be expressed in terms of с?,
from B6.9), as in the coherent case. When the result, ef, is substituted into
B6.10), the six components of efj are found to be linear functions of a. The
unknown a is then determined from condition B6.11), while the associated
elastic strain energy can be determined from B5.3).
Lee and Johnson A978) have calculated W*/V for the spheroidal precipi-
precipitate (a1 = a2 = a, a3/a = j8), where the crystallographic directions of the
precipitate and the matrix are parallel to the principal axes of the spheroid.
Their result is
W*/V= \i2K*c°/(K* + c°), B6.12)
where K* = \{C%X + 2Cf2), which is the bulk modulus of the precipitate, and
i + SU22 - 1)A - S3333) + 2SU33S3311})
+ 3 E*3333 ~~ ^1133 ~ ^ззп ~Ч~ 2^1133^3311) B6.13)
with K= j(Cu + 2C12). C* and CtJ are the elastic moduli of cubic crystals.
The Eshelby tensor Sijkl has been given by A7.27) and A7.28), where /? = l/p.
Kroner A954) has denoted by c° the compression modulus. Figure 26.5 shows
the numerical results of W*/VC44e2 calculated by Lee and Johnson A978).
The elastic strain energy reaches its maximum when the precipitate is a sphere,
vanishing as the shape becomes a thin, flat ellipsoid.

228
Chap. 4 Ellipsoidal inhomogeneities
0 30
025 -
020 -
T 0.15 -
1 ' I T T 1 Г
Си
l /^~
' К KCI
- 1/ Си matrix
i i i i i i
i
Сл
AL
КС]
010 -
005
0 0 2 0.4 0.6 0.8 10 1.2 1.4 oo
/8
Fig 26 5 Normalized elastic strain energy per unit volume of the incoherent precipitate vs aspect
ratios for the Cu precipitate, Al precipitate and KCI precipitate in a Cu matrix
The incoherent state of the precipitates characterized by B6.10) and B6.11)
can be interpreted as a free-energy minimum state of coherent precipitates
with given efj, when an additional inelastic deformation Aefj is allowable*
Consider a coherent precipitate with eigenstrain efj. The corresponding stres*
field is denoted by atj and is calculated from B6.9). Let us consider a
relaxation which occurs inside or at the surface of the coherent precipitate.
Dislocations punching out of a precipitate represent the relaxation process
occurring outside, which may be caused by actual plastic deformation, by
mass diffusion along the matrix-precipitate interface, or through the bulk of
the precipitate as suggested by Christian A965). This inelastic deformation
due to relaxation is denoted by Atfj. Either case of diffusion or plastic
deformation does not cause volume changes,
4^ = 0. B6.13.1)
The total stress in the precipitate is a^ + Aaijb where Aatj is the eigenstress due
to Aefj. The associated elastic strain energy W* is
B6.13.2)
from B5.3).

26. Precipitates and martensites
229
The minimum principle of the free energy W* is introduced. Aef, is created
so that W* becomes minimum for a given efj. This extreme condition leads to
B6.13.3)
since a,y + Aa^ is a linear function of cfj + Aefj, which is the solution of B6.9)
when efj is replaced by e? + Aefj. Condition B6.13.3) is satisfied when
because of B6.13.1); therefore, W* becomes
B6.13.4)
B6.13.5)
Condition B6.13.4) is no more than B6.10); that is, the situation considered
above is identical to that of an incoherent precipitate.
Martensitic transformation
One of the important applications of the inclusion theory is in the area of
martensitic transformation. Although various aspects of martensitic transfor-
transformation have been investigated (Bilby and Christian 1956, Christian 1965,
Wayman 1964, etc.), not until quite recently have the details of the application
of the inclusion theory appeared in the literature.
The problem to be discussed here concerns the elastic field and elastic
strain energy associated with the martensitic transformation. We assume that
Fig. 26.6. Disk-shaped martensite blade, a1 — a2 = a, fi = a^/a

230
Chap. 4 Ellipsoidal inhomogeneities
the orientation of martensite blades relative to the crystalline directions is
determined by the principle of minimum elastic strain energy.
Mura, Mon, and Kato A976) have considered a thin disk-shaped marten-
site blade (flj = a2 = a, /J = аъ/а <s: 1) as shown in Fig. 26.6. The x coordi-
coordinate system is chosen parallel to certain crystallographic directions of the
matrix, while the x' coordinate system coincides with the principal axes of the
ellipsoid. In many ferrous alloys, martensites take the form of a thin, flat
ellipsoid which consists of alternative twins, as shown schematically in Fig.
26.6. Each twin can be considered to be formed by the respective transforma-
transformation strains (eigenstrains). In the case of martensitic transformation from a
face-centered cubic to a body-centered tetragonal, the eigenstrain components
in the x coordinate system are given by
P —
?l о о
О е2 О
о о «п
2 ° °
е1 °
0 0 с,
\ 1 /
in region I,
in region 2,
B6.14)
and (see e.g. Wechsler, Lieberman and Read 1953)
B6.15}
where b0 is the lattice constant for the f.c.c. phase and b and с are those for
the b.c.t. phase. Region 1 is denoted by hatching in Fig. 26.6.
The Fourier series expression of B6.14) is
B6.16)
where
= B/L) fL/2 ef cosBw«jc;/L) dxj.
•'-L/2
B6.17)

26. Precipitates and martensites 231
L is the wave-length of the alternating twins. «^@) is the mean value of tfj. If
the volume ratio of regions 1 and 2 is denoted by /, it holds that
€&@) =Нг + A-/K, B6-18)
The eigenstrain components in the x' coordinate system are given by
<?/ = "*,¦<*//<?/. B6-19)
where аъ- is the direction cosine between the xk- and the x,'-axes.
As a first approximation, let us start with the case where е?у = ё? @). This is
similar to the approximation adopted by Shibata and Ono A975, 1977) and
Easterling and Tholen A976). Equation B6.7) can be used for the elastic field.
When
e{V = €?r = ef2, = 0, B6.20)
the stress components vanish everywhere. Therefore, the associated elastic
strain energy becomes zero. By solving B6.18) with B6.19), it is found that
condition B6.20) can be satisfied if
a3y = cose=(-el/t2)l/2, a13, = (l+flA2I/2, a2y = 0, B6.21)
and
/=<i/(«i-e2). B6.21.1)
where в is denoted in Fig. 26.6.
In the phenomenological crystallographic theory of martensitic transforma-
transformation, the deformation in the martensite is assumed to be uniform, correspond-
corresponding to the first term in B6.16). Furthermore, the phenomenological theory
assumes that the magnitude and the angle between any two vectors lying
parallel to the flat interface do not change (Wayman 1968). This is equivalent
to B6.20). The postulates in the phenomenological crystallographic theory are
interpreted in such a manner that the elastic strain energy vanishes when the
uniform transformation is assumed. Thus, the prediction of / and the orienta-
orientation of the martensite plate based on the phenomenological theory should
agree with B6.21) and B6.21.1). In fact, this is the case when | e, |, | e2 | «: 1

232 Chap. 4 Ellipsoidal inhomogeneities
(Wechsler, Lieberman, and Read 1953). The last restriction is that of the small
deformation theory, while the calculations in the phenomenological theory are
based on the finite deformation theory.
When we consider the effect of higher order terms in B6.16), the equivalent
inclusion condition B2.11) is written as
% = C*k,(ikl-€l,) = CiJkl(ekl- ?]f; -e*,), B6.22)
where e*j is the equivalent eigenstrain. The Fourier series expression is
assumed for «**,
<*/ = «7/@) + Ё ??;(«) cosilvnxt/L), B6.23)
where
e**=c? + c*. B6.24)
From A7.67), condition B6.22) is written as
Л-1
- E «*/*(«) cosBiriixi/Z,)}, B6.25)
n-l )
where | is the unit vector along the Xj-direction. When the orientation of the

26. Precipitates and martensites 233
flat ellipsoid and / are chosen as defined by B6.21) and B6.21.1), the first two
terms on both sides of B6.25) disappear by taking
Г/ v4/—?Г/\^/' \ZO.Zo)
Further simplification of B6.25) can be made when the terms containing Jo
are neglected. Observation by electron microscope has shown that a ~ 10 /xm
and L = 100 A (e.g., Tamura et al. 1964). Thus, JQ{2iran/L) = J0{<x>) ~ 0.
Finally, equation B6.25) becomes
- C,%, ? ё?,(и) cos{2vnx[/L) = -Cijkl ? i*kf(n) cosBwnxi/L) B6.27)
и-1 «=-1
which is also equal to atJ. Therefore, from B6.16),
atj = ~ c*ki {eki - «7/@)} • B6.28)
The elastic strain energy is
W*=-\\aijf.fj AD, B6.29)
Jy
where V is the volume of the flat ellipsoid. When B6.14) and B6.18) are used
in the above two formulae, we have
W*/V = f(l -f){C^ - C*2)(el -t2) , B6.30)
where / is given by B6.21.1). This is surprisingly large, presumably because a
very thin, flat ellipsoid is assumed in the calculation.
Suezawa A978) has used the method of Khachaturyan A967) and evaluated
the strain energy of martensitic transformation to be proportional to (L/a3J
(see also a review paper by Christian 1979).
Shibata and Ono A975, 1977) have considered the martensitic transforma-
transformation of titanium. They assume elastic isotropy for both the matrix and the
martensite. A homogeneous spheroidal inclusion (a1 = a3 = a, k = a2/a) is
subjected to a uniform transformation strain efj (eigenstrain). The elastic
strain energy is expressed by A3.20.1), where e* is replaced by ef,,, (see Fig.
26.7).
The transformation of the b.c.c. phase of Ti and Zr alloys during quenching
is martensitic and results in at least four structures: h.c.p., f.c.c, orthorhombic
and f.c.c. orthorhombic (Williams 1973). The h.c.p. martensite is most com-

234
Chap. 4 Ellipsoidal inhomogeneities
Fig. 26.7. Spheroidal martensite at = 03 — о, к = а2/а.
mon and occurs in Ti, Zr, Ti-Mo alloys, Zr-Nb alloys and others. In pure
metals and other dilute alloys, the lath martensite forms, whereas the plate
martensite occurs with increasing solute content. The orientation relationship
of the plate martensite to the matrix has been established by Burgers A934),
(Oil) B || @001) „,
[111] в || [1120] я.
By taking the xx-, x2-, and ^-directions as
respectively, Shibata and Ono A977) have used
/ 0.0381 -0.0884 0 \
ifj=\ -0.0884 -0.0434 0
1 0 0 0.0088/
B6.31)
[211]B and [011]s,
B6.32)
where the ratios of the lattice parameters in b.c.c. and h.c.p. phases are taken
as aH/aB = 0.899 and cH/aH= 1.587. Denoting the principal axes of the
spheroid by x'x, x'2 and x'3, we have the relation B6.19). The rotated coordi-
coordinate system x' is obtained by rotating first about the x3-axis by в, followed by
the rotation about the x[-axis by \p (see Fig. 26.7). For a given еЛ, the elastic
strain energy per unit volume of martensite becomes a function of в, -ф and
к = a2/a. Shibata and Ono have calculated в, \р and k, under which the
elastic strain energy attains its minimum value. Their results indicate that the
elastic strain energy is minimized when the morphology of h.c.p. Ti martensite
is a thin disk shape lying on a plane close to A1Л')В, where X is equal to
1 • 2 ~ 1 • 3. This habit plane agrees with experimental observations.
A spherical Fe precipitate, which is formed in a Cu matrix and is f.c.c.
before martensitic transformation, becomes b.c.c. with the banded structure of
alternating twins after transformation (Easterling and Miekk-oja, 1967). The
elastic strain energy of an Fe particle with the banded structure has recently

26. Precipitates and martensites
235
/ ^ \
/M = l IM = 2 \M
Fig 26 8 Spherical precipitate undergoing martensitic transformation When matrix constraint is
absent, the spherical particle changes its shape successively to (a), (b) and (c) to form twinned
martensite
been evaluated by Kinsman, Sprys, and Asaro A975) using an unrealistic
approximation based on a simplified model. The physical situation they have
envisaged is a spherical region undergoing martensitic transformation, with the
upper and the lower halves shearing in anti-parallel directions, as shown
schematically in Fig. 26.8(a). Thus, the surface and the eigenstrain of the
transformed particle J2 are
= ± i
B6.33)
where a is the radius of the spherical particle and S is a constant to be
determined by the crystallographic information. However, Kinsman, Sprys,
and Asaro A975) approximated B6.33) with the eigenstrain
c$1(x)=tx3/a in Q,
B6.34)
with t = S, in order to utilize Asaro and Barnett's work A975). Mura, Mori
and Kato A976) have calculated the elastic strain energy associated with
situation B6.33) as follows: The Fourier series expression for the eigenstrain
defined by B6.33) is
= BS/*r) ? Bи-1) 1sin{Bn-lO7x3/a}.
B6.35)
For simplicity, assume that the spherical precipitate and the matrix have the

236 Chap. 4 Ellipsoidal inhomogeneities
same isotropic elastic constants. From A7.56), the only non-vanishing stress
component a31 pertinent to the strain energy evaluation is
00
*31(*)= I*>i(*. «). B6-36)
n = 1
with
o3l(x, n) = (pSa3/irC3) I |/o(Coi?) sin{(x • ?)C3i3/a)
+ 2J0(C0R)C3i3(x ¦ i) cos{(x • i)С3C/а}/a
_i_ 7 (Г* J?\ ( ( v ?\^/^ /^/i2d\ t r2?2p //^ \ c;« Г/* -«. ?\f"* ? /« \ I
x Ш + п - ЪШ) dS@ - (fyS/Сэ) sm(C3x3/a), B6.37)
where C3 = B« - l)w, /x is the shear modulus, and Poisson's ratio is taken to
be |. Since the analytical integration involved in B6.37) is difficult, the stress
and the elastic strain energy to be determined from B6.29) have been
evaluated numerically on a computer. The result is
W*/V=0.1%2nS2. B6.38)
The above value is a little larger than that of Kinsman, Sprys, and Asaro
A975).
Similar calculations have been performed for the cases where
, . I \S, 2Na/M<x3<BN + l)a/M )
*%i(x) = , ч ,ч in 0, B6.39)
\-\S, BN+l)a/M<x3<BN+2)a/Mj
with M = 2, 3 (see Figs. 26.8(b), (c)). (The case M = 1 corresponds to B6.33).)
In B6.39) N takes on integer values. The calculated elastic strain energies are
plotted against M in Fig. 26.9, which also includes results for the case M = 0,
corresponding to uniform shear and the value of Kinsman, Sprys, and Asaro
A975) for the case M = 1. It can be seen that as M increases, the elastic strain
energy decreases. Thus, as far as the elastic strain energy is concerned, the
alternating twin structure with the shorter interval is favored. This conclusion
agrees with intuition, based on an examination of Fig. 26.8; as the interval

26. Precipitates and martensites
237
<\J
>
—
nerg
ш
и
о
Ш
0.6
0.4
0.2
-
_
-
ч
4
N
N
X
4
0
•
_
Present study
Kinsman et al.
~ o-
M
Fig 26 9 Elastic strain energy of a martensite particle with M pairs of twins in Fig 26 8
between the alternating twins decreases, the shape change of the particle
caused by the transformation becomes smaller as a whole.
Stress orienting precipitation
As a typical example, we discuss the precipitation of a" phases, Fe16N2, from
a supersaturated Fe-N alloy. The eigenstrain (misfit) of this precipitate has
the components
ep = €p = -0.0023, e{3 = 0.0971
B6.40)
(Jack, 1951). There are two more types of the crystallographically equivalent
Fe16N2, eigenstrains of which are obtained by the cyclic permutation of the
index in B6.40). The type characterized by B6.40) is called the @01) variant,
since the precipitates having B6.40) are formed in disk shapes, approximately
parallel to the @01) plane. This can be understood by noting that | efx | =
I ?22 I ^ I ?зз I an(i tnat *е elastic energy of the disk-shaped precipitate is
determined by the eigenstrains within the disk plane (see A7.42), B6.7) and
the related discussions).
Statistically, the three types of the variants, A00), @10), and @01), are
formed in an equal amount, when there is no external or internal stress, except
the stress due to the precipitates. However, when an external stress is applied,
a particular type of variant is preferentially found; e.g., the presence of the
tensile stress along [001] favors the formation of the @01) variants against
others (Nakada, Leslie and Chray, 1967). This phenomenon can be interpreted
by the interaction between the precipitates and the applied stress; the interac-
interaction energy B5.24) is far smaller for the @01) variant than for the A00) or

238
Chap. 4 Ellipsoidal inhomogeneities
@10) variant, when the tensile stress is applied along the [001]. This effect,
known as the stress orienting effect, is remarkable, and it has been shown that
as the tensile stress increases, not only the relative number but also the
absolute number of the @01) variants becomes larger (Tanaka, Sato and Mori,
1978).
Sato, Mori and Tanaka A979) examined the precipitation orientations
around a non-metallic inclusion subjected to a tensile stress as shown in Fig.
26.10. In the figure, the tensile stress a303 = 98 MN/m2 has been applied along
the [001] (vertical) direction during the precipitation of Fe16N2 in a Fe-N alloy
single crystal. It is noted that most of the precipitates are parallel to the @01)
plane. With a close examination it can be seen that the density of the
precipitates is larger in the close neighborhood of the inclusion (black con-
contrasted spheroid) along the [010] (horizontal) direction and is small at the top
or the bottom of the inclusion, the [001] direction. Sato, Mori, and Tanaka
have calculated the stress outside the inclusion from A8.6) together with the
method of the equivalent inclusion in Section 22. The elastic constants are
Cuu = 2.37 X 10" N/m2, CU22 = 1.41 X 1011 N/m2 and C1212 = 1.16 X 1011
N/m2 for Fe. Since the non-metallic inclusion is identified as a slug particle,
the isotropic elastic constants are assigned to the inclusion (the shear modulus
Fig 26 10 Precipitates around a non-metallic inclusion subjected to a tensile stress (after Sato,
Mon, and Tanaka 1979)

26. Precipitates and martensites
239
»\
Fig 26 11 Equi-contour lines of the stress disturbance around an inclusion (after Sato, Mori, and
Tanaka 1979)
2.9 X 1010 N/m2 and the Poisson ratio 0.22, for typical glass). Figure 26.11
shows the equi-contour lines of the stress disturbance which is expressed in
units of aA. In agreement with Fig. 26.10 the stress at the [010] edges is
significantly large while that at the [001] edges is smaller than aA.

240 Chap. 5 Cracks
Cracks
An elliptical crack is a special case of an ellipsoidal void where one of the
principal axes of the ellipsoid becomes vanishingly small. The Griffith fracture
criterion and stress intensity factors of cracks are the primary topics discussed
in this chapter.
27. Critical stresses of cracks in isorropic media
Penny-shaped cracks
Cracks can be considered as a special case of voids when one of the principal
axes, say a3, is infinitesimally small compared to the others. The simplest
example, a penny-shaped crack (ax = a2 = a, a3 = 0), is considered first. As-
Assume that an infinitely extended isotropic material containing a crack J2 is
subjected to a uniform applied stress a303 at infinity. The equivalent inclusion
method is employed for the analysis.
The crack is simulated by a penny-shaped inclusion with an eigenstrain efj.
The corresponding eigenstress a,y is given by A1.26). The necessary conditions
for the application of the equivalent inclusion method are
< + °* = ° in<2 B7.1)
a31 = CT32 = 0
which correspond to traction-free conditions on the crack surface. It can be
seen in A1.26) that no finite values of c*j satisfy the first condition in B7.1)
for a3 —> 0. Therefore, we allow €33 to be unbounded in such a manner that
lim a3e?3 = finite = e*. B7.2)
<23->0

2 7. Critical stresses of cracks in isotropic media 241
All other components of t? are taken to be zero. Thus,
JU.7T ?*
C33~~2(l-p)~^' B7.3)
<*31 = а32 = 0-
Conditions B7.1) are satisfied if e* is chosen as
^20-Of.^ B74)
The interaction energy (the decrease in the total potential energy of the body
due to the crack) is given by B5.18),
B7-5)
The Griffith A921) fracture criterion is
~(AW+2'na2y)=0, B7.6)
where lira2 is the total surface of the crack and у is the surface energy per
unit area. Condition B7.6) leads to the critical stress
B7.7)
A given penny-shaped crack with radius a becomes unstable (starts to grow)
when the applied stress a303 reaches the critical value given by B7.7).
A penny-shaped crack under a uniform shear 03° at infinity can be treated
in a similar manner. All components of t*} except i*x are taken to be zero in
A1.27), where the unbounded «5i is sucn
lim a3t^ = finite = e*. B7.8)
a,->0

242 Chap. 5 Cracks
We then have
^{2v) ^ . .
°31~ 2A-,) a' B7-9)
The traction-free conditions on the crack surface are
<+°"=°0 A7.10)
CT33 - a32 - U>
which are satisfied by
The interaction energy thus becomes
AW= -DV3)a2a3V=-83(/x1(;!);K«J, B7.12)
and condition B7.6) leads to the critical stress
°» \ 2A -,)«
Results B7.7) and B7.13) have been obtained by Sack A946) and Sneddon
A946). The interaction energy and the crack opening displacement for a pair
of coplaner penny-shaped cracks are given by Fu and Keer A969) for the
shear case and by Fu A973) for thermal cracks.
Slit-like cracks
When one of the principal axes of an ellipsoidal void, say a3, becomes infinite
and the minor axis a2 becomes infimtely small, the ellipsoid becomes a
slit-like crack. When the unbounded solid which contains this crack is sub-
subjected to a uniform tension a22 at infinity, the stress disturbance due to the
crack can be simulated by A1.22.1). In A1.22.1) we choose all с*, = О except
the unbounded t%2 which must satisfy
= finite = e* fora2->0. B7.14)

27. Critical stresses of cracks in isotropic media 243
Then we have
B7.15)
1 1 - v a1
The traction-free conditions on the crack surface are
<4 + °22 = 0, B?16)
°21 ~ a23 - 0,
which are satisfied when
t' = (l-v)aii&/p. B7.17)
The interaction energy per unit length of the crack in the x3-direction becomes
AW= — ?7гага 2O22t 22 = —jira-iO^*
= -v(l-v)al(o?2f/2li, B7.18)
and the Griffith fracture criterion
OCl-i
leads to
(AW+4a1y)=0 B7.19)
When the body containing this slit crack is subjected to a uniform applied
stress a°2 at infinity, a similar analysis leads to
I A. \l/2
p'2=\gA_t,)fl / • B7.21)
The above result, B7.20) and B7.21), is for the plane strain case. The
corresponding result for the plane stress is obtained by replacing E/(l — v2)

244 Chap. 5 Cracks
by E and v/{l — v) by v. Then, we have the critical stresses
. B7.22)
The above results have been obtained by Griffith A921).
Flat ellipsoidal cracks
When a crack is a flat ellipsoid (ax > a2, аъ = 0), the stress disturbance can be
simulated by A1.26) or A1.27) depending on the applied stress.
For simple tension a303, we have
\im a3t*33 = o?3a2(l - р)/цЕ(к),
a3-»0
B7.23)
AW= -2rn{o^1
where k1 = 1 - a\/a\, and E(k) and F(k) in B7.24) are the complete elliptic
integrals defined by A1.25).
For simple shear o31, we have
B7.24)
a\-a\
— v
The expressions for AW in B7.23) and B7.24) have been obtained by Kassir
and Sih A967).
The critical stress can be obtained from
О Ui
3 B7.25)
1Г-(А1?+2тта1а2у) = 0,
OCI2
I2
if ax and a2 are regarded as independent variables. The following formulae

2 7. Critical stresses of cracks in isotropic media 245
are useful for calculation of B7.25):
dE(k)/dk={E(k)-F(k)}/k,
dF(k)/dk= [E{k)/{\ - k2) - F(k)}/k, B7.26)
дк/даг = а\/а\к, Ък/Ъа2 = -а2/а\к.
For simple tension, the first equation in B7.25) yields
^2E2(k) Г B7.27)
(l-v)a2[(l-2(kf)E(k)
and the second equation gives
3p,yk2E2(k)
B7.27.1)
l(l-v)a2[{2-(k>J)E(k)-(k'JF(k)}
where
k2 = l-al/a2, (k'J = aj/a2. B7.27.2)
The solution of the simultaneous equations B7.25) leads to ax = a2 and a303
given by B7.7). For a given set of аг and a2, where ax Ф a2, the critical value
of а3°3 is given as the smaller of B7.27) or B7.27.1). When аг > а2, the smaller
value is B7.27.1). The crack grows from the minor axis of the ellipse. This is
consistent with the fact that the stress intensity factor at the minor axis is
greater than that at the major axis.
For simple shear, the first equation in B7.25) yields
3jLtY
(l-v)a2
-|V2
{A - v) - C + v)(kf + 2(k'L}E(k) +{Bv+ l){k'f - (k'L}F{k)
B7.28)

246 Chap. 5 Cracks
and the second equation gives
4 =
{\-v)a2
X i
{2 - 2v + Df - 3)(/t') + (?')}?¦(?) + {(? - l)(/t') + A - 3v)(k')}F(k
B7.28.1)
When 1 > а2/аг > 0.7, (p = 0.3), 03° given by B7.28) is smaller than o^ given
by B7.28.1). Therefore, the crack starts to grow from the major axis of the
ellipse. If, however, a2/ax < 0.7, 0,° given by B7.28.1) becomes smaller than
a30! given by B7.28). Then, the crack starts to grow from the minor axis. When
ai/a\ > I» expressions B7.28) and B7.28.1) must be rewritten by using the
following formulae:
= {a2/a,)E{k),
= (ai/a2)F(k),
= l-a2/al, B7.28.2)
= F/(F-i),
Smith A971) has investigated the case when 8аг and Sa2 satisfy the
condition w = 8а2/8аг. Не considered a penny-shaped crack which in its
initial state is subjected to an applied stress field a^ and o°j. This penny-shaped
crack may grow into an elliptical crack. The Griffith fracture criterion may be
obtained from B7.23) and B7.24) as
v a2{F(k)-E(k)} E(k)
^ + —
+ 2waia2y\=0, B7.29)

27. Critical stresses of cracks in isotropic media 247
where
[^^Ь- B7'30)
The limits a1 -»a and a2—>a are taken in B7.29), since the crack is initially
penny-shaped. Then, B7.29) and B7.30) lead to
B7.31)
B-pf(l+m)
The right-hand side of B7.31) is a maximum at m = 0. It implies that for a
penny-shaped crack with a given size the minimum a°3 is obtained when
m = 0 under a constant a31, or the minimum o^ is obtained when m = 0 under
a constant a303. Therefore, it can be seen that a penny-shaped crack under a3l
and a303 grows into an ellipse, and further, that there is no growth perpendicu-
perpendicular to the direction of the applied shear stress. Generally it is believed that a
penny-shaped crack extends to another penny-shaped crack under the action
of a°3 alone. According to Smith's theory, a penny-shaped crack extends to an
elliptic crack under a°3 when a non-zero a3Oj is also acting.
Crack opening displacement
Since a crack is regarded as an ellipsoidal void by letting a3 -»0, the crack
opening displacement is obtained by equating the two expressions for the
interaction energy B5.17) and B5.18.6),
j AD = hjsa?jnj[u,] dS. B7.32)
The volume element dD in fl is written as 2h dS, where h is the half-thick-
half-thickness of the ellipsoidal void in the ^-direction and
h = a,(\-x\/a\-x\/a\). B7.33)
Thus, we have
-a°€*2A = a/»,] B7.34)
since B7.32) holds for any 2. For ст° = а303, B7.34) gives [и3] = 2Ле$3 and

248 Chap. 5 Cracks
[щ] = [u2] = 0. For a° = a30, (i = 1 or 2), B7.34) gives [u,.] = 4A«?,. and [и3] = О.
The normal on the crack и, is defined on the upper surface before the
opening. a3e* are given by B7.4), B7.11), and B7.17). This method is
applicable for anisotropic materials.
28. Critical stresses of cracks in anisotropic media
Uniform applied stresses
Consider an elliptic crack {x\/a\ + x\/a\ = 1) in an anisotropic medium. The
x coordinate axes are chosen parallel to the principal axes of the ellipsoid,
x\/a\ + x\/a\ + x\/a\ = 1. The elliptic crack is degenerated from the el-
ellipsoid by taking a3 -* 0. Let <j° be the applied stress at infinity.
The boundary conditions on the crack surface are
a303 + or33 = 0,
а3°+а31 = 0, B8.1)
а302 + а32 = 0,
where а,; is the stress disturbance due to the crack. This stress disturbance is
simulated by the eigenstress derived from an equivalent ellipsoidal inclusion
with eigenstram ef7. From A7.12) we have
B8-2)
where
B8.3)
Since we are interested in the limiting case a3 -* 0, the integral in B8.2) is
modified. As shown in A7.42), a3j becomes zero for this case when ef, is finite.
However, a3j becomes nonzero if ef, is infinite and a3cf, is finite for a3 -* 0.
Nonzero values of a,y are necessary in order to satisfy the boundary conditions
B8.1).

28 Critical stresses of cracks in anisotropic media 249
It holds that
Introducing the following transformation:
( }
and since Gkpl4(?) is a polynomial of degree zero, we express B8.4) as
л, йв r» Gkplq{oo&e,smB,t)At
L ~TT1) -—:—~ :——.—:—гтттт;- С28-6)
•'о a^a1a-i J-<x>
Bв/a\al + %v^ f
When integration by parts with respect to / is performed, using
~{t/(cos2e/ala2 + sin26/a2a2 + t2/afaljl/2}
= (cos2e/aja2 + sin2e/aja2)
X (cos2e/ala2 + sm2e/alal + t2/ala22)~V1 B8.7)
and
(() ,0,l), B8.8)

250 Chap. 5 Cracks
B8.6) becomes
4) a1a2(cosle/aZ + sin0/ajf)
dGkplq(cos в, sin 0, f )/3f d?
¦ B8.9)
Г/2
°° (cos0/at + sin2e/at)(cos2e/al + sin^/a? + alt2/a\a\)
Noting that
йв
г = 2тахаг B8.10)
f2v
/
•'о
/^ + /?
and that a3 <c d^, a2>we finally have
a,y= (l/4w)CVJt/q,,mjJ?*J,{4wGA;>/,@, 0, 1) - а3ПкрЫ} - CIJklt*kl, B8.11)
where
_ /2^ d<? / kp/g( в, sin 0, 0/Э? d/
/2' dfl r» Gg^/g(cos g, sin в, t) ^
Jo a\a\ J-~ (a-22cos20 + ef2 sin20)V2
after integrating by parts, where G^^ is chosen so that [tG1plq]'tZ^x = 0 and
Gkplq — G%plq = constant, namely, the quotient of two polynomials of degree 6
divided by a polynomial of degree 6 is rewritten in a constant plus a
polynomial of degree 4 divided by a polynomial of degree 6. The constant is
ignored when the integration by parts is performed. Gkplq = Gfplq when / and
q are not 3 at the same time.
As has been shown already by A7.42), the first and third terms in B8.11)
cancel each other out on the crack surface. The boundary conditions B8.1)
therefore become
o°j-(l/4TT)C3jklCpqmna3e*mnIIkpl4 = 0 0 = 1,2,3) B8.13)

28 Critical stresses of cracks in anisotropic media
251
Fig 28 1. The x' coordinate system coincides with the crystallographic directions of a crystal.
or
where
Lijmn = ~
= 1,2,3),
B8.14)
B8.15)
The set of three equations B8.14) can be solved for the remaining eigenstrains.
Mura and S.C. Lin A974) and Cheng A977) have considered B8.14) in
some detail when the crack is oriented as shown in Fig. 28.1, where the x2-axis
coincides with one of the crystallographic directions, say the Xj-axis. The other
two crystallographic directions are taken as the x[- and *3-axes.
For a simple tension case (a3° = a303, normal to the crack surface), they have
found that
«3**3 = - <L3131/A, a3e*31 = a303?3133/24, B8.16)
and for a simple shear case (а3° = о31, tangential to the crack surface),
a3e?3 = a3°1L3331/4, а3€?г = -a3°L3333/2A, B8.17)
where
A = ?3333Ь3131 ~ -?-3133-'
B8.18)

252
Chap. 5 Cracks
~ 100
S
30"
ISO"
60' 90" 120"
a (in degree)
Fig. 28.2. The critical stress in simple tension.
All other components of ef, can be taken as zero. It is found that L3133 = 0,
L3331 = 0 for a = 0.
The interaction energy is
B8.19)
For a penny-shaped crack (ax = a2 = a), the critical stress is obtained from
B8.20)
where у is the surface energy. Numerical results obtained by Mura and S.C.
Lin A974) and Cheng A977) are shown in Fig. 28.2 and Fig. 28.3 for simple
tension and shear, respectively. Two materials, iron and zinc, are examined. In
the case of zinc, the x\, x^-plane is taken as the basal plane. The physical
constants are taken from the American Institute of Physics Handbook A972).
For iron, X = 14.1, fi = 11.6, and ju' = -13.6 with units of 1011 dyne/cm2, and
у = 2.0 X 103 dyne/cm. For zinc, Cn = 16.1, C33 = 6.1, C^ = 3.83, C12 = 3.42,
and C13 = 5.01 with units of 10й dyne/cm2, and у = 708 dyne/cm. It is
interesting to observe that the critical a3°3 of iron takes a minimum value at

28. Critical stresses of cracks in anisotropic media
253
9 0 -
80
Zinc
О* 30* 60° 90° 120° 150° 1Э0"
а (in degree)
Fig. 28.3. The critical stress in simple shear.
а = 90°, while the corresponding quantity for zinc obtains its maximum at
а = 90°. The critical value of o^ for iron reaches a maximum value at
а = 90°, but that of zinc takes a minimum value at a = 90°.
Non-uniform applied stresses
If the applied stress is given in the form of
the eigenstrain used in the equivalent inclusion method becomes
B8.21)
B8-22)
The eigenstress inside the equivalent inclusion ?2 can be obtained from A9.12),
where S* = S2. When a3 <§: аъ а2, we have, as seen in A9.17),
aij = Cijk, { Cpqmncsmn(xs/as)Gkplq{0, 0, 1) - f.pklxp/a
p/ap
B8.23)

254 Chap. 5 Cracks
where
f
n
L,jmn V/^)CijklCpqmn 5/2 1/
s (ai cos 0 + Oj sard) A - ?3)
B8.24)
and
fl = Olll, f2 = ^2^2> ?з = 0>
? = (cos 0, sin 0,0),
ii = (l-f32I/2cos0, B8.25)
12= {l-ЦI/2 sine,
The boundary conditions are
4 + a3j = 0 U = 1.2,3). B8.26)
It can also be shown that the first two terms in B8.23) cancel each other for
o3j. However, the above boundary conditions can only be satisfied by taking
a3esmn = finite for a3 -»0. The applied stress B8.21) cannot be given arbi-
arbitrarily. Since a°jj = 0 must be satisfied, we restrict attention to the case where
аз1! = аъ\ = а|3 = 0. B8.27)
For simplicity, the crack plane is assumed to be one of the crystallographic
planes (a = 0 in Fig. 28.1). Then, for simple tension, a303 = a^x^/a-^ + a33x2/a2,
„ A _ __1 /TU
аЗ?33~ ^ЗЗ/^ЗЗЗЗ. /^ .m
n 2 _ „2 /,22 B8-28^
"з?зз~ "зз/^зззз»
and for simple shear, аъ\ = оъ\х2/а2,
a3e2n=-c3\/Lful. B8.29)

28. Critical stresses of cracks in anisotropic media 255
If the material is isotropic,
Ьзззз = -ii{(k'JF(k) + Bk2 - \)E{k)}/{\ - v)a2k\
L23233 = ,x{(k'JF(k) - (k2 + \)E{k)}/(\ - v)a2k2, B8.30)
?3131 = /*{(k'JF(k) - (k2 + \)E(k)}/a2k2,
where k2 = l-aj/a2, (к')г = a\/a\, and E(k) and F(k) are defined by
A1.25).
* IT integrals for a penny-shaped crack (a1 = a2)
The integrals defined by B8.12) are given below for cubic and hexagonal
crystals when the crystallographic directions coincide with the x coordinate
system (a = 0 in Fig. 28.1). The crack plane is the basal plane for hexagonal
crystals.
* П integrals for cubic crystals
iI2222 —
ааг Jo
2тг ri
ааг Jo
p(p + q)
1л2 + кх2
РЯA-х2I/2
aX
\V2
dx-
ааг Jo
-— /1
аах Jo |
4тг /-1
a^i ^o
.2„4/'1 2\1/2,
7Г ,1[12х4A-х2I/2(с-2Ь-Зсх2)
dx
2(к-2м2) к-:
МA-х2I/2
\V2
Vi ,.2\l/2
(pqY(l-x2)
\!/2
dx
aax

256
IT,,,, =.
Chap. 5 Cracks
dx
277
dx-
aa-.
n _JL
iI22U "" __
р(р + ч)
2ет /-1
аах Jo
jU2 + КХ2 _
pq{l~x2f/2 „
dx
\l/2
dx-
B8.31)
g Г1
7г /-ix4(l— x2) (ju.2 + кх2)(с — 2b — 3cx2)
dx
2ba
27Г /-1
(8к-у)х4+(У-к)х2
\V2
dx
Х + М) /-1
aax Jo
Ол,

28. Critical stresses of cracks in anisotropic media
fi*2(l-*2I/2
Пшз-П2323-
where
dx
1/2
bx2(l-x2)]1/2.
П integrals for transversely isotropic materials
257
B8.32)
•111 ~ ^2222
2aJ0
* ri / [ /A - x2) + hx2] [(d+3e)(l - x2) + 4/x2] - g2x2(l - x
J
4I/2
\V2
2Д77

258
Chap. 5 Cracks
= - — Aex2(l - x2I/\d-4fx2) dx
a\ Jo [
i\Afx4(d-4fx2) f3h(d-4f)
A-х2?/2
\V2
- g2 - 2/7.
J2(l-xf/2fh
/2fh2
|
W2 /2 - g2 - 2fh + dh
2211
* ril\f(l
2ax Jo \
4 | , 2tti/2
ax —
V^A(l-xI
:, B8.33)
— ^
2233

28. Critical stresses of cracks in anisotropic media 259
2W2 Bf2 + hd+he-g2)
«1 f2h
2477 д fi\ Ax61 1 л , 24W2
4[ 1/2J /
(f2 + dh- g2)x2(l - x2) +fhx4} + [e(l - x2) +fx2]
X[(f2-2fd+dh-g2)(l-x2)
*!. I1
8тг fi3fh-dh-eh-f + g
~f~ i . ._ ax
J fo~1/2
aj
- x2) +fx2}
where A =
^)+fe2]-g2x2(l-^)}, B8.34)
and ^, h, f, g, and e are defined in A7.30).

260 Chap. 5 Cracks
29. Stress intensity factors for a flat ellipsoidal crack
Stress intensity factors of cracks as well as critical stresses are important
quantities in fracture mechanics, as has been pointed out by Irwin A948). The
stress intensity factors are also important in the study of crack growth rates of
fatigue cracks in connection with the Paris law A964). Stress fields around
cracks and their stress intensity factors are well documented in the literature,
e.g., Sneddon and Lowengrub A969), Sneddon A961), Irwin A958), Sih and
Liebowitz A968), Paris and Sih A964), and Kassir and Sih A975), among
others. However, little is known about the stress intensity factors of materials
which are anisotropic or when crack shapes are arbitrary.
In this section we follow the work of Willis A968) on the stress field around
a flat ellipsoidal crack in general anisotropic elastic media. The applied stress,
o°, is assumed to be a polynomial function of the coordinates. The x
coordinate system is chosen as shown in Fig. 28.1. The stress disturbance is
denoted by a,y. The total stress field is <т° + atj. The displacement disturbance
м, is related to a,- • through
o,j=CijklukJ. B9.1)
The boundary conditions are
of3°. + 03|. = O when*3 = 0, xl/al + x\/a2<\,
ut{x) -> 0 when | x | -> oo
where a30, is a given polynomial function of хг and x2.
It can be expected that the displacement field м, and the corresponding
stress otJ have properties similar to those derived from eigenstrains ef,. From
A6.2) we have
Xexp{-i{-(x-x')}d?dx'. B9.3)
When
e*3m(x') dx' = -bm(x') dS3(x') = bm(x') dx[ dx'2 B9.4)

29. Stress intensity factors for a flat ellipsoidal crack 261
is defined on the crack surface, B9.3) becomes
xexp{-ig-(x-x')} й?йх[йх'2, B9.5)
where S = S3 is the crack surface and b is called the Burgers vector of the
Somigliana dislocation A914). Eshelby A957) has found that | b | is propor-
proportional to A — x\/a\ - x\/a\)l/1 when the applied stress at infinity is uni-
uniform. In the present case where the applied stress is polynomial, b(x) is
chosen as
b(x)=l(Xl/ai + ix2/a2Y{xx/ax - ix2/a2L(l - x\/a\ - x\/a\)X/\
B9.6)
where A is a constant vector of a complex number which will be determined
later together with p and q, and p>q. The ^-integration can be performed
by using Cauchy's theorem. Then, B9.5) becomes
JJs
Xexp{-itN-(x-x')}dx{<lx2, B9.7)
where
y-duii.tUSLii)) B9.8)
and ^3 = |^(|1; ?2) are tne roots °f tne equation
?>(?)= 0 B9.9)
with negative imaginary parts for xi > 0. For isotropic materials there is only
one double root with a negative imaginary part (N — 1). For a double root the
residual is obtained from 3{(€,{/Л'Л/,(«)/)-1(€Ж«з - Й1J}/^-
The stress components are
„ { \ ~i V ff^ At At CUklCpgm^g^Nk (?N ) , f
4тг n=1->J~cX> dD(?")/dti3 JsJ
Xexp{-HN-(x-x')} dx{djc^. B9.10)

262
Chap 5 Cracks
The functions ?^(f 1( |2) are homogeneous of degree 1. It should be noted that
Nkp{?) and D(?) are homogeneous of degrees 4 and 6, respectively.
The complicated integrations in B9.7) and B9.10), when b(x) is given by
B9.6). are carried out by Willis A968). The results are
:-i)-
-ът[ъr.
Jn
3
^=1
ах, r\2/a2,
v 9 /-{(g+DAg-l)}172
v'g2 -
X
X Я +
X
w
p*\
B9.11)
— /-277
X
-¦m1/2
X
1
w
X
B9.12)

29. Stress intensity factors for a flat ellipsoidal crack
where
=x1/a1, y2 = x2/a2, y=
When x3 = 0,
)
263
B9.13)
B9.14)
where
Р Я
Е Е (-i)("
m + n
m + n
B9.15)
*.p + q — m — n
B9.16)
It is seen that ajJ(x1, x2, 0) is a polynomial of degree (p + q) in yx, y2 when
|j>| < 1. The boundary conditions B9.2) are satisfied by choosing suitable
constants Ъъ Ъ2, Ъ3, when a30, is a polynomial of degree (p + q) in уг, у2.
The relative displacement of the two crack surfaces is
1, x2, -0)=bi(x1, x2).
B9.17)
This is based upon the fact that the Burgers vector b (or dislocation density on
a plane) is defined by the multiple-valuedness of the displacement across the

264 Chap. 5 Cracks
crack plane. Since the displacement is symmetric about the crack plane,
ut{xx,x2, +0)= \bt{xx, x2). When Ъ is determined by the boundary condi-
conditions, the elastic field is completely determined by B9.11) and B9.12).
The interaction energy B5.17) becomes
*mbm{x')<ix'ydx'2 B9.18)
which is used for the Griffith fracture criterion calculation.
In B9.16) we have Km = 1, Ku = Hv -уJ + i, Kw = (vy), and K21
— iD ¦J'K + hiv 'У)- These values are used for a constant applied stress
p = q = 0 and a linear applied stress p = 1, q = 0.
Uniform applied stresses
The integral with respect to w in B9.12) is denoted by /. Then, for p = q = 0,
W^JLt^-jL. B9Л9)
3 2 Я 1 1
Since the imaginary part of g is positive, — w < arg(g + l)/(g - 1) < 0. When
x3 = 0 and >> < 1, we have g = - (?) • y) and log(g 4- l)/(g - 1) = log | A +
?)/(! - g) I -'«¦¦ Furthermore, log |A + g)/(l - g) | and g/(g2 - 1) are odd
functions of t) for jc3 = 0. On the other hand, Y?N=\Fijm is an even function of
t,. Therefore, for x3 = 0 and у < 1, - |log |A + g)/(l - g) \ and g/(g2 - 1)
do not contribute to B9.12) and we have
°u= ~b Г ? Fum{vi/ai, Чг/аг, йзЫМ, V2/a2))bm йф. B9.20)
The boundary conditions B9.2) determine the unknown constants Ъх, Ъ2, and
The stress intensity factor at a crack tip is defined by
kt]= lira Bт7гI/2а,7, B9.21)
where r is the normal distance from the edge of the crack. For a flat
ellipsoidal crack,
r=(y- l)/(x2/a< + x\/a\f\ B9.22)

29. Stress intensity factors for a flat ellipsoidal crack
265
The stress components atJ in B9.21) can be evaluated from B9.12) in the
neighborhood of the crack edge. When л: is close to the crack edge, the second
term in B9.19) is dominant. Accordingly, we consider the integral
B9-23)
where g = - (tj • y). We have
B9.24)
g - 1 " •'О f-1
where y-y/y. The first term in B9.24) remains bounded as у -* 1, so that
0A).
B9.25)
Thus,
О ДГ.1
E
¦o(i)
B9.26)
The stress intensity factor B9.21) becomes
X
fl?, x2/a22,
Ai2, ^2A22))bm.
B9.27)
The stress intensity factor of a flat ellipsoidal crack in transversely isotropic
media has been investigated by Shield A951), Kassir and Sih A968) and
Hoenig A978). Their methods differ from the Willis approach A968). Explicit
expressions for kx, ku, and kul are presented by Hoenig.

266 Chap. 5 Cracks
For isotropic materials, B9.9) has two roots. The root with a negative
imaginary part is
*3($i,f2)=-'-Ui2 + €!I/2. B9.28)
Using C.34), we can write B9.13) as
= /<32iV?i. «г. h) X + 2u— 2fXT~T ^ 2' '
= ^323 = ^33i = ^332 = 0, B9.29)
and B9.20) becomes
Q\a2 2 \ + » alP{vl/a2
If o° = a3°3, boundary conditions B9.2) yield Ьх = Ьг = 0, and
B9.31)
is defined by A1.25). Result B9.31) is consistent with B7.23), since

29 Stress intensity factors for a flat ellipsoidal crack 267
Ъъ. The stress intensity factor кзъ in B9.27) is denoted by кг for
к, = о°га2^(х1/а* + x22/a2I/4/?(A:). B9.32)
For a penny-shaped crack where ax = a2 = a and E(k) = \tr,
l/1 B9.32.1)
The result B9.32) has been obtained by Sadowsky and Sternberg A949) while
the special case of a penny-shaped crack has been considered by Sneddon
A946).
In a similar manner, we obtain the stress intensity factor for shear stresses.
When o,° = 03°, boundary conditions B9.2) and B9.30) lead to
- v)E{k) + v(l - k*)F(k)} B9.33)
and Ь2 = Ь3 = О, where F(k) and E{k) are defined in A1.25). The stress
intensity factors B9.27) become
B9.34)
B9.34.1)
Denoting the normal vector along the ellipse by v, we define the stress
intensity factor of the second kind ku by
*ii = *3i"i+*32> B9.35)
where

268 Chap. 5 Cracks
From this, we have
{k2-v)E{k) + v{\~k2)F{k) V '
In the derivation, the following integrals are used:
.1/2
B9-38)
\ a{ aj j
Expression B9.37) is consistent with Kassir and Sih's result A966).
Non-uniform applied stresses
Let us consider the case where the applied stress before the disturbance due to
the crack is a linear function of хг and x2. Then, p = 1 and q = 0 are chosen
in B9.6). When the integral with respect to w in B9.12) is denoted by /, we
have
For x3 = 0 and y<\, /(g g| g)/g)|
g log |A + g)/(l - g) | and B-3 g2)/(g2 - 1) are even functions of tj. Since
e'* is an odd function of tj, B9.12) becomes
3/
''•J= ~~I f " ^ ри>«Ы/а1' Чг/а2, ${щ/а\, Vi/a2))
4 Jo лг=1
X(r,l + irJ)bm(i]-y)u<j>. B9.40)
The boundary conditions B9.2) determine the unknown constants.
For a stress near the crack edge, the second term in B9.39) is dominant.

29. Stress intensity factors for a flat ellipsoidal crack
Thus, for x3 = 0 and у -> 1,
3 ~)
U 2w ^0 лг = 1 '¦"" m g2-l
269
+ 0A). B9.41)
The stress intensity factor becomes
X ? FiJm{Xl/ai, x2/a\, tf{Xl/al x2/a\))
If the material is isotropic and the applied stress is given by
B9.42)
B9.43)
we choose
b3 = b + ib',
where b and b' are real numbers. Then, condition a33
B9.40) leads to
B9.44)
-a303 applied to
B9.45)

270 Chap. 5 Cracks
The stress intensity factor кгъ is denoted by kY,
B9.46)
where
/2i
Ei
(k) = r/2sin2<>(l - A:2 sin^)V2 ёф,
•'о
/2 - k2 sin^)V2 ёф, B9.47)
The stress intensity factor kn defined by B9.35) is derived in a similar
manner, while the stress intensity factor km of the third kind is defined by
fcme*3i'i+*32'2. B9.48)
where vector t is the tangential unit vector along the ellipse.
Kassir and Sih A968) have found that k{ under a uniform a°3 is indepen-
independent of the elastic moduli of a transversely isotropic solid. Barnett and Asaro
A972) have found that for a slit-like crack the stress intensity factor Ktj is
independent of the elastic moduli for any arbitrarily anisotropic material.
Sekine and Mura A979) have modified Willis' method A968) and proved
that if the displacement discontinuity of the elliptical Somigliana dislocation
A914) has the form PN(xv *2)A -x\/a\ - x2/a2I/2, where PN is a homo-
homogeneous polynomial of degree N, the stresses on the plane of the Somigliana
dislocation are inhomogeneous polynomials of the coordinates, whose terms
are of degree N, (N — 2), (N - 4), The same subject has been discussed by
Kassir and Sih A967) under the more restrictive condition PN = PN(x2, x\).
Certain special cases relating to elliptical disk-shaped cracks have been
examined in the literature. Chen A966) has solved the problem of an elliptical
crack subjected to linearly varying loads in a transversely isotropic body.
Stress intensity factors for an embedded elliptical crack or a semi-elliptical
crack, which approach the free surface of the semi-infinite solid under uniform
tension perpendicular to the plane of crack, have been investigated by
Miyamoto and Miyoshi A971), Shah and Kobayashi A973), and Nisitani and
Murakami A974). No analytical expressions are available for the stress inten-
intensity factors.

30. Stress intensity factors for a slit-like crack
111
30. Stress intensity factors for a slit-like crack
The stress intensity factors of a slit-like crack in anisotropic media can be
obtained from B9.27) and B9.42) as a special case when ax —» oo and
xi ~* ai= a- However, more explicit expressions can be obtained by the
method of continuously distributed dislocations (Bilby and Eshelby 1968). We
follow the work of Barnett and Asaro A972) in this section.
Consider a slit-like crack in an infinite anisotropic elastic medium (Fig.
30.1). afj denotes the applied stress at infinity. The stress disturbance induced
by the crack is denoted by
boundary conditions are
°2°i + °2< = ° for x2 = 0, |
so that the total stress field is
atJ.
The
с
C0.1)
ou is simulated by the stress field due to continuously distributed dislocations
on x2 = 0, | xx | < c, where the dislocation line direction is parallel to the
x3-axis. From the elastic theory of dislocations (Willis 1970 and Barnett and
Swanger 1971), the stress at xx, x2 = 0, due to a single dislocation located at
x1 = t, x2 = 0 with the Burgers vector bj, is expressed as
o2i = K,jbJ/(x1-t). C0.2)
The boundary conditions C0.1) may be satisfied if the dislocations are
continuously distnbuted with strength bj(t) (Burgers vector times distribution
density). Then, C0.1) becomes
K
4f
•* — r
C0.3)
Fig. 30.1. Slit-like crack with length 2 c.

i = К и
272 Chap. 5 Cracks
or
f b)(t)(x,-tyldt=-K-,lo°,. C0.4)
J -c
The inverse matrix components Kjt' are given by
тг{->^КтгКП!./21аЬ<.К.ХаК1ЬК.Ъс, C0.5)
so that
К/Кп = КлК~1=8л. C0.6)
Equation C0.4) is the Hilbert integral equation for bj(t).
Uniform applied stresses
For constant a20,, as seen in Appendix 4, the solution of C0.4) is easily
obtained as
bJ(t) = KJl^ltMc2-t2I/\ C0.7)
where Qo in (A.4.5) of Appendix 4 is taken to be zero in view of condition
f bj(t)dt = Q. C0.8)
This condition is satisfied, since there is no relative displacement of the crack
surface at x, = ±c. The relative displacement (crack opening displacement)
Auj at xx is given by
AUj
= fbj(t)dt C0.9)
J v.
from the dislocation theory.
The stress traction acting on the plane of the crack (| xl \ > c, x2 = 0)
becomes
(\Xl\>c). C0.10)

30. Stress intensity factors for a slit-like crack 273
The stress intensity factor defined by B9.21) leads to
k2i = a2°,VwF C0.11)
where г = (хг — с). The last equation is Barnett and Asaro's remarkable result
A972). The stress intensity factor k2i due to a2i is independent of elastic
moduli. Result C0.11) is a special case of B9.32) and B9.34) for ax -» oo,
a2 -* с
From C0.7) and C0.9), the crack opening displacement is
4«, = *;>°(<2-*i2I/2A. C0.12)
The interaction energy AW is expressed from B5.18.6) as
AW = -\( o20\AUj dxj C0.13)
which becomes
AW=-\K-y2i4]C\ C0.14)
The critical applied stress can be obtained by the use of B7.19) where ax = c,
and
-d(AW)/dc = \Kj№i°2jc- C0-15)
When the material is isotropic, Kn = K22 = ju./2w(l — v), K33 = jx/Itt and
Ko = 0 (/ *j) (see Read, 1953). The cntical stresses obtained from C0.15) and
B7.19) agree with B7.20) and B7.21).
The tensor components appearing in C0.14) and C0.15) are referred to the
x coordinate system shown in Fig. 30.1. Since the form of C0.14) and C0.15)
is invariant of coordinate systems, we can choose a new coordinate system in
the crystallographic directions in which the elastic moduli appear in their
simplest form.
The crack opening displacement near the crack tip is obtained from C0.12)
with C0.11). For an isotropic material, K22 = /x/2w(l - v) and (c2 - x\)l/1«
BcrI/2, where r is the distance from the crack tip, and therefore
[u2] = 4*,A - v){r/2m)l/2/ii. C0.16)
This result is applicable to any plane crack since the crack opening displace-
displacement near the crack edge is a local quantity. On the other hand, [u,-] is

274 Chap. 5 Cracks
obtained from B7.34). When the two expressions for the crack opening
displacement are put to be equal, we have an equation to determine the stress
intensity factor.
An example is shown for an elliptic crack. From B7.34) and B7.4) we have
[u3] = 2а3<?зA - x\/a\ - x22/a22I/2
= 2a2(l - »H3O3A - x\/a\ - xl/alI/2/iiE(k)
= 2a2{\ - p)aU2rI/2{x2/at + x\/a\f"/v.E{k), C0.17)
where r is the normal distance from the crack edge defined by B9.22). On the
other hand, we have from C0.16)
[и3]=Лк1A-р)(г/2ттI/2/ц C0.18)
where the normal direction to the crack surface is taken as the x3-direction.
Equating C0.17) and C0.18), we have
kl = а2^о°3{х2/а* + x2/a42I/A/E{k) C0.19)
which is identical to B9.32).
Non -uniform applied stresses
When an applied stress a2i is a function of xx on x2 = 0, | хг | < c, the integral
equation C0.4) still holds. The solution has been obtained by Barnett and
Asaro A972). Since the involved analysis is essentially the same as that in the
following section for the Dugdale crack, the calculation is not reported here.
Isotropic materials
The stress intensity factors of a slit-like crack in an infinitely extended
isotropic material are the same as C0.11). These factors, k22 = klt k21 = kn,
and k23 = klu, have been obtained by Westergaard A939). Irwin A948)
pointed out the relation between the critical stress and the stress intensity
factor (see also Irwin 1957, 1958). The associated stress intensity factors for
cases where an infinitesimally thin notch (crack) is introduced into a semi-in-
semi-infinite slab have been investigated by many researchers (see Wigglesworth 1957,
Lachenbruch 1961, Koiter 1965, Nisitani and Murakami 1972, and others).

30. Stress intensity factors for a slit-like crack 275
For a strip with a center notch (crack), numerical calculations for the stress
intensity factors have been performed by Isida A971). The stress intensity
factor for an inner or edge crack in arbitrarily shaped plates has been
investigated by Murakami A978) by using the Green function for the
traction-free crack in an infinite plate (see also Erdogan 1962). Yamamoto and
Sumi A978) recently reported that a slit-like crack is not a two-dimensional
problem (neither plane strain nor plane stress), since for moderate values of
the ratio of crack length to the plate thickness and because of the effect of
Poisson's ratio, the necessary conditions for neither plane stress nor plane
strain can be satisfied. They evaluated the variation of the stress intensity
factor along the crack front.
Stress intensity factors of notches and cracks in various standard test
specimens and machine elements under various applied loads have been
investigated by a number of researchers, and many of the results are listed in a
handbook by Tada A973) and in a special publication by Kamei and Yokobori
A974).
A recent development is the use of the finite element method in the
evaluation of the stress intensity factors (see Zienkiewicz, 1971). Early original
work in this area is found in papers by Watwood A969), Chan, Tuba and
Wilson A970), and Dixson and Pook A969). Since the stress field in the
neighborhood of the crack tip is proportional to \/\fr (r is the distance from
the crack tip), the proportionality constant, i.e., the stress intensity factor, can
be obtained by plotting the finite element solution as a function r. The stress
intensity factor can also be found by plotting the crack opening displacement
versus r. These two methods are called the stress and displacement methods. A
third method, an energy method, has been proposed, based on the fact that the
energy release rate due to the crack extension is proportional to the square of
the stress intensity factor. The first two methods do not give satisfactory
results unless very fine meshes are employed, while moderate accuracy can be
obtained by the energy method. The energy method, however, cannot be
applied to mixed mode or three-dimensional problems. Recently, the stress
method has been reconsidered by Miyata and Kusumoto A975) and Murakami
A976, 1978). They have found empirical relations between stress intensity
factors and stress solutions obtained by the finite element method, and they
have obtained stress intensity factors for several three-dimensional cracks with
relatively coarse meshes.
Byskov A970), Rao, Raju, and Krishna Murty A971) and Pian, Tong, and
Luk A971) proposed special elements by using the hybrid vibrational method
or the collocation method. These methods introduce additional freedom in
conjunction with Williams' coefficients A957) for stress singularities. Another
interesting approach, proposed by Yamamoto A971), is illustrated in Fig. 30.2.

276
Chap. 5 Cracks
Fig. 30.2. Superposition method.
The exact stress field a is a « a" — ar + ae, where a" is an analytical solution
for an infinite body, ar is a solution by the finite element method in the finite
body subjected to a boundary force with the same traction as at the corre-
corresponding imaginary boundary in an infinite body, and ae is a solution of the
finite body with the originally given load by the finite element method.
Yamamoto postulated that, in the neighborhood of the crack tip, a = a(aa —
ar) + ae = aaa, where a is introduced as a Lagrangian multiplier which
coincides with the stress intensity factor. Thus, we have a = ae/ar. This
method has been further extended by Yamamoto and Sumi A978, 1979) to
include three-dimensional crack problems.
The theory of stability of interacting tension cracks in brittle solids has
been recently developed by Nemat-Nasser A978), Nemat-Nasser, Keer, and
Parihar A978), Keer, Nemat-Nasser, and Oranratnachai A979), Sumi,
Nemat-Nasser, and Keer A980), and Horii and Nemat-Nasser A986), in
connection with thermally induced edge cracks in plane strain conditions.
These authors show that as a half-space is cooled on its free surface, tension
cracks develop and penetrate the half-space. However, a critical state is then
reached at which some cracks stop, as others grow at a faster rate. Moreover,
depending on the cooling mechanism, another critical state may be attained at
which the cracks which had stopped may actually snap closed as the remaining
ones jump into a finitely longer length. The growth process is highly imperfec-
imperfection sensitive, as shown by Nemat-Nasser and Shokooh A980). In a series of
experiments on glass plates, Geyer and Nemat-Nasser A981) obtained good
qualitative and quantitative verification of the theory.

31. Stress concentration factors
31. Stress concentration factors
277
Let us consider an ellipsoidal inhomogeneity in an infinite medium (see Fig.
31.1). afj denotes the applied stress, and otJ the stress disturbance due to the
inhomogeneity. аи can be simulated by the eigenstress associated with eigen-
strain efj of an equivalent inclusion in a homogeneous medium. The stress
immediately outside the inclusion can be written as
where
[°l7]=<TG(out)-al7(in);
C1.1)
C1.2)
see F.11). The equivalent eigenstrain ef, in F.11) must be determined from the
equivalency condition B2.7).
The stress concentration factor к of afj for a given o°, = 5 is defined by
If the inhomogeneity is a void, cr,y(iii) + ofj = 0, and
C1.3)
C1-4)
Mura and Cheng A977) have calculated к for a three-dimensional, lens-shaped
crack, where a3 is much smaller than аг or a2- Since a3 ¦« ax, a2, the results
B8.16) and B8.17) can be used. For simplicity the case a = 0 in Fig. 28.1 is
Fig. 31.1. A lens-shaped crack.

278
Chap. 5 Cracks
considered. The definition C1.3) does not mean that к is the same for all
stress components.
Simple tension
Let a303 be prescribed with all other components of a° being zero. Combining
C1.4), F.11), and B8.16), we have the stress concentration factor,
K= D77/аз){С3333 ~ C33klCpg33Gkplq{n)}/C33mnCst33^msn,' C1-5)
where Gkplq{n) = Nkp(n)n,nq/D{n) and Umsnt is defined by B8.12). The
interesting points to be investigated are on the crack boundary with x3 = 0.
The radius p at a point corresponding to angle jS in Fig. 31.1 is
1/2
P = {al/a2){sin2ii + (fla/fliJ cos2y8} .
Equation C1.5) is then written as
К =
1/2
С 77
nKjst33xl
sin2/?+ -^
«1
1/4
C1.6)
C1.7)
2.0 -
1.0
0.5
10
a, /a2
100
Fig. 31.2. Maximum tensile stress concentration factor divided by (a2/p)l/1 for various ratios of
ах/аг.

31. Stress concentration factors 279
For isotropic materials it becomes
/ 2 \1
= 2(a2/p)l/2 sin20+^cos2)8
I ai /
C1.8)
where E(k) is defined in A1.25). When к is considered at the point /? = 90°
with a1 -* oo, it agrees with the well-known solution given by Inglis A913).
Numerical values of к/(а2/рI/2 at /? = 90°, where the stress concentration
factor becomes maximum, are plotted in Fig. 31.2.
Pure shear
Let Oj" be prescribed with all other components of a° being zero. From
C1.4), F.11), B8.17), and C1.6) the stress concentration factor of a^ is
_ 4тг \Qi3i ~ CnklCpqnGkplqyn) j I a\_ 2/?| /o-i Q\
For isotropic materials and /? = 90°, it becomes
2}j, C1.10)
where F(k) is defined in A1.25) and k'1 = 1 - k2.
Numerical values of к/(а2/рI/2 at fi = 90° are plotted in Fig. 31.3. The
elastic moduli in Figs. 31.2 and 31.3 are taken from the American Institute of
Physics Handbook A972). It is observed that when аг/а2 approaches infinity,
the values of к/(а2/рI/2 approach constants which agree with the values
obtained from Lekhnitskii's method A968). The resulting displacement field
agrees with Chen's solution A966) for an elliptical disk-shaped crack in
transversely isotropic materials.
The state of stress due to an ellipsoidal cavity in an infinite isotropic
medium has been given by Sadowsky and Sternberg A949) and Miyamoto
A953) for the case of three constant normal stresses, applied at infinity,
having their directions aligned with the principal axes of the ellipsoid. Earlier
solutions obtained by Neuber A937) consider loading in applied shear as well
as normal stresses for the special case of the spheroidal cavity. Mirandy and
Paul A976) also employ the equivalent inclusion method. They investigate the
effect of Poisson's ratio on the stress concentration factors of an ellipsoidal
cavity subjected to a completely arbitrary constant state of stress at infinity.

280
i.o
Q8
V Q6
0.4
02
Chap. 5 Cracks
Mg
Zn
10
100
Fig. 31 3. Maximum shear stress concentration factor divided by (а2/р)х/г for various ratios of
ах/аг
It should be pointed out that the stress intensity factors generally cannot be
derived by the limiting process
kx — limy/2 тгр ко®.
C1.11)
32. Dugdale-Barenblatt cracks
In the absence of plastic deformations around crack tips (or notches), some
stress components become infinite there. In reality, however, the stress field
Fig. 32.1. Dugdale-Barenblatt crack.

32. Dugdale-Barenblatt cracks
281
Perfectly elasto-plastic material
Work - hardening material
Fig. 32.2. Stress-strain curves.
cannot exceed a certain limit because of the plastic deformation in the
neighborhood of the crack tips. Dugdale A960) has simpUfied the situation
and has assumed that the plastic region is one-dimensional, being confined on
(c, a) and ( — a, — c), as shown in Fig. 32.1. He has related the size of the
plastic domain to the applied stress and crack length 2 c by assuming the
material to be elastic-perfectly plastic (Fig. 32.2a). One method of coping with
this problem of stress singularity is that of Barenblatt A960). He postulates a
system of cohesive stresses in the corresponding region (a, c) and (-a, — c)
in Fig. 32.1. The magnitude of these stresses is just sufficient to cause the usual
stress singularity to vanish.
The boundary conditions for a Dugdale crack under a simple tension,
a202 = S, are
O2j =
o22 =
o22 =
И2 = 1
0
k0
S
U = 1>
1=0
2)
at
at
at
at
x2
x2
x2
x2
= 0
= 0
= oo,
= 0
|.
с
a
xi
<
<
1 <c,
\x1\<a,
l*il.
C2.1)
where a22 is the applied tensile stress and k0 is the tensile yield stress. The

282 Chap. 5 Cracks
following stress analysis shows that an = a22 = o}3 for the plane strain and
an = ст22, ff33 = 0 f°r the plane stress at the crack tips. The boundary condition
ff22 = ^o is therefore valid only for the plane stress assumption, since a22 = k0
comes from the maximum shear theory \{a12 — a33) = \kQ.
The differential equations to be solved are, for the isotropic materials,
= 0. C2.2)
When the Papkovich-Neuber potential ^ is used,
2<zMl=-— #д-х2*12,
1 + к C23)
where
*,n + *,22 = 0, C2.4)
к = C - f )/(l + >») for the plane stress, and к = 3 - 4? for the plane strain,
where p is Poisson's ratio. The stress components derived from C2.3) are
a22= ^,22 — х2^222,
. C2.5)
Before we apply C2.5) to C2.1), it is convenient to rewrite C2.1) as
<J22
°n
°22
a22
= 0
= 0
= — s
= k0- S
at
at
at
at
x2
x2
x2
= 00,
= oo,
= 0,
= 0,
с <
1 <c
l*il
<a
\ ¦)
at x2 = 0,
The solution for C2.1) is obtained from the solution for C2.6) by superim-
superimposing S onto a22.
From the symmetry of the problem with respect to хъ Ф is chosen as
~ix* cos ?*! d|, C2.7)

32. Dugdale-Barenblatt cracks 283
which satisfies C2.4). The first three conditions in C2.6) are automatically
satisfied. With C2.5), the fourth and fifth conditions yield
C2.8)
о
where
ct(xJ = 5 for | л:г | <c
= S-/c0 forc<|Xi]<a. C2.9)
Assuming
A(t)= fq(t)J0(tt)dt, C2.10)
where /0 is the Bessel function of zero order, from C2.3) and C2.7) we have,
at x2 = 0,
т^-и2= fq(t) dtf°J0(tt) cos
1 + К Jq Jo
= / ?@— —T77 d?
= 0 forfl<|x1|. C2.11)
The last condition in C2.6) is thereby satisfied. The unknown function q is
determined from C2.8). Substituting C2.10) into C2.8), we obtain
4 Г ^ иг dt f«l*il<e- C2Л2)

284 Chap. 5 Cracks
Integrating with respect to xl we have Abel's integral equation
where
P(x1)= po(s)ds. C2.13.1)
•'o
•'o
The solution is
g(,),±jLj-~i-v~"—' C2.13.2)
or, integrating by parts,
- dXl. C2.14)
When a(xl) is given as C2.9), C2.14) becomes
q{t) = St ioxt<c,
iovc<t<a.
q(t)t{(SkQ)cos+Ssm
it \ t
All quantities of the elastic field can be obtained from C2.15) by using
C2.7) and C2.10). In order to have a finite value of stress, however, we must
have
<jr(a)=O, C2.16)
because, at x2 = 0, a < \ x: |, we have
, . #(*! -t) , d ra , л , . _! t
— a

32. Dugdale-Barenblatt cracks
From C2.15) and C2.16) we have
cos
-l
¦n_S_
a 2 k0'
285
C2.17)
The above equation gives the relationship between the applied stress and the
plastic zone size. The relation has been found by Dugdale A960).
Of particular interest is the displacement component u2 in the plastic
domain. It is obtained from C2.11) and C2.15) for x2 = 0+:
и2(хъ c) =
к)к0
(хг + с) cosh г
m
C +
— (x1 — c) cosh
-l
m
с-хл
log:
¦-cW-xlI
\V2
\V2
+ c log-
\V2
C2.18)
where m = (a2 - c2)/a and n — с/a. The value at x2 = 0 is the negative of
C2.18). The displacement at the upper tip of the crack is obtained from
C2.18) by setting хг = с,
W, =
277JU.
coshW-(- + -\\.
\a с.
C2.19)
When a/c = 1 and S/k0 «: 1, C2.17) and C2.19) are approximated as
l-c/a=T(S/k0)\
(l + *)fco n . ч
l - c/a) =
u2=
2-7ТЦ
1 + k) S2
— етс-j—,
16ц A:o
C2.20)

286 Chap. 5 Cracks
and we have
u2(Xl, c)/u2(c, c) = (l-|I/2-flog^(|~^, C2.20.1)
where ? = (хг — c)/(a — c).
Orowan A945,1949) and Irwin A948,1957) have proposed that the applied
critical stress S required to propagate a crack of length 2 c in a semi-brittle
solid of elastic constant ц should be written in the form of a Griffith equation,
i.e.,
S^(n/cI/2, C2.21)
in which у represents the sum of the true surface energy and the plastic work
required to increase the area of the crack by a unit amount. This Orowan-Irwin
formula can be obtained from C2.20). We postulate that the crack becomes
unstable when the crack opening displacement 2u2 given in C2.20) reaches a
critical value Ф. Then,
The strain hardening effect can be taken into account in this result if an
approximation is employed as done by Theocaris and Gdoutos A974). Alter-
Alternative interpretations of C2.21) are given by Weertman A978) and Vilmann
and Mura A979).
A similar result can be obtained for the anti-plane problem. A shear stress
a302 = 5 is applied at infinity. The boundary conditions are
a32 = S at x2 = oo,
a-.-, = 0 at x-, = 0, I x-i I < c,
C2 23)
a32 = k0 at x2 = 0, c<\xx\<a, \ ¦ /
w = 0 at x2 = 0, a<\x1\,
where w is the displacement in the x3-direction. Again we rewrite C2.23) as
a32 = 0 at x2 = oo,
a32=-5 at,2-0, \Xl\<c
a32 = k0 — S at x2 = 0, c<\x1\<a,
w = 0 at x2 = 0, a -< | jcx |.

32. Dugdale-Barenblatt cracks 287
The solution for C2.23) is obtained from the solution for C2.24) by super-
superimposing S onto 0. Hooke's law yields a31 = ц-wл and a32 = juu'2, and the
equation of equilibrium a31l + a322 = 0 leads to
w,n + wj22 = 0. C2.25)
We assume w has the form
w = (l/n)jA(Z)e-tX2 cos fcxx d?. C2.26)
Then,
«32 = - ГиЦ)е-*Х1 cos ix, d? C2.27)
'o
which satisfies the first condition in C2.24). The second and third conditions
in C2.24) yield
-OO
/ $Л($) cos **!<!$ = a(x,)> C2.28)
where
o(x,) = S for \xr\<c, C2 29)
a(xl) = S — k0 for с < \хг | < a.
The above result is the same as C2.8) with C2.9). Taking
q(t)J0(it)dt, C2.30)
we have, from C2.15),
q(t) = St for t<c,
2 ( с с \
q(t) = —tl{S- k0) cos - + S sin - } for с < t < a.
it \ t t)
C2.31)

288 Chap. 5 Cracks
w at x2 = 0 is calculated from C2.26) and C2.11) as
/" a(t)
dt ioi\xl\<a,
-~^-x\)V2 C2.32)
[iw = 0 for a < |xj |.
Substituting C2.31), we have, similar to C2.18),
ITfl
т
c + x
— (x1 — c) cosh
-i
m
с- хл
C2.33)
where m = (a2 — c2)/a and n = c/a.
The condition for the finite value of a32 at xx = a is C2.16); therefore,
When a/c =¦ 1 and 5/A:0 «: 1, C2.34) and w at Xj = с are written approxi-
approximately as
2 C2.35)
w--fco(e-c).
We can see from this result that the critical stress for the anti-plane problem
also has the form given by C2.21).
BCS model
Bilby, Cottrell and Swinden A963) have solved the anti-plane problem given
by C2.23) or C2.24) using a completely different method. The stress dis-
disturbance caused by the crack and plastic region is simulated by the dislocation
stress. The dislocations are distributed continuously in the domain ( — a, a).
The idea that cracks can be simulated by arrayed dislocations has originated
in the work of Eshelby A957).

32. Dugdale-Barenblatt cracks
289
Fig 32 3 Distnbution of dislocations.
Consider a crack as shown in Fig. 32.3. a203 = 5 is an applied stress. The
dislocation stress a23 at x2 = 0 due to a screw dislocation at xx = t, x2 = 0 is
([ib/2'ir)/(xl — t). The boundary conditions corresponding to C2.23) can be
reduced to the Hilbert integral equation
S+ f" (ti/2^)b(t)(x,-ty1dt={°
J-a \K0
for \xy\ <c,
for с < | хг | < a,
C2.36)
where b(t) is the strength of the Burgers vector (Burgers vector times the
density of the dislocation distribution). As seen in Appendix 4, the work of
Head and Louat A955) leads to the solution for C2.36),
for
C2.37)
provided that condition (A.4.7) is satisfied. This condition gives (A.4.9) or
c/a = cos(TrS/2k0) C2.38)
which coincides with C2.34).

290 Chap. 5 Cracks
The relative displacement Au3 at x2 = 0 (crack opening displacement in
| x11 < с and plastic displacement in с < | х: | < a) is obtained from
Au3 = f"b(t)dt. C2.39)
. JXl
It becomes
2kJ Xl(gi-ciI/2-c(qi-xlI/2
Л1
2 2
-c
C2.40)
The relative displacement at xx — с is obtained by the limiting process,
—г, becoming
Ли3 = Dко/кц)с log(fl/c). C2.41)
For small plastic zones, (a — c)/c «; 1, C2.38) and C2.41) become, approxi-
approximately,
C2.42)
Аи3 = {Лко/щ){а - с) = itcS /2\ik0,
which agree with C2.35) where Au3 = 2w. The critical stress is similar to
C2.22). The distribution of dislocations C2.37) is sketched in Fig. 32.3.
The BCS model has been extended by many researchers. In order to
investigate plastic yielding from sharp notches, Bilby et al. and A964) Bilby
and Swinden A965) have considered an infinite array of relaxed cracks.
Fracture characteristics of solids containing doubly periodic arrays of cracks
have been investigated by Kanhaloo A978, 1979).
Barnett and Asaro A972) have considered the BCS model in an arbitrarily
anisotropic elastic medium. As shown by C0.2), the dislocation stress at

32. Dugdale-Barenblatt cracks 291
x1 = хг, x2-0 due to a single dislocation located at xY = t, x2 = 0 with the
Burgers vector b is
a2l. = X,yV(*i-0- C2.43)
Assuming a continuous distribution of dislocations with the strength b(t), we
write the boundary conditions given in the domain x2 = 0, | хг | < a, as
JT0.6,@(*i ' О"' d/ = 0 for | x, | < c,
C2.44)
a20, + / КиЬ^1)(Х1 - /) d? = fc,(jd) for с < | X! | < a,
•>-a
where a20, (i = 1, 2, 3) are components of the applied stress at infinity and /c, is
the component of yielding shear stress. It is convenient to rewrite C2.44) as
f bJ(t)(x1-tyldt = Kj(ki(x1)-o»i), C2.45)
J —a
where /сДл^) = О for \хг\ < с When fc, is constant, the solution of C2.45) has
the same form as (A.4.8), where k is replaced by KJ^kj.
The BCS model has been further extended to cracks with inclined slip
planes (Fig. 32.4) by Atkinson and Kay A971), Vitek A976), Riedel A976),
Atkinson and Kanninen A977), Karihaloo A979), Takeuchi A979), Miyamoto
et al. A981), Hayashi and Nemat-Nasser A981), and Chang and Ohr A984). A
further extension to the two-dimensional dislocation distribution is presented
in Section 41.
The Dugdale model has been used to obtain a dynamic solution by Goodier
and Field A963) and Kanninen et al. A968). The dislocation distribution
predicted by the BCS model was observed by an electron microscope by
Kobayashi and Ohr A979) and Ohr and Narayan A980).
In the Dugdale-Barenblatt or BCS crack, the stress at the crack tip has no
singularity. On the other hand, Thomson A976, 1978) and Weertman A978,
1980) have proposed a crack model in which a small enclave that is sur-
surrounded by a plastic region is considered to exist at the crack tip. This small
enclave is a dislocation-free zone and the stress at the crack tip has the same
singularity as the elastic crack. The stress intensity factor, however, is shielded
by the dislocation distribution in the surrounding plastic zone. Further exten-
extensions of the theory have been made by Chang and Ohr A985) who considered
inclined dislocation emission, by Li A985) who numerically calculated the
interaction between a crack and dislocations to investigate the rate of disloca-

292
Chap. 5 Cracks
Fig. 32.4. Crack with inclined slip plane.
tion emission during loading and unloading, and by Weertman A983, 1984)
and Weertman, Lin, and Thomson A983) who proposed a double-slip plane
model to obtain crack growth equations for Mode II or III under a monotoni-
cally increasing stress.
Penny-shaped crack
Let us consider a penny-shaped crack with radius с as shown in Fig. 32.5. The
material is assumed to be isotropic and infinitely extended. When a uniform
Fig. 32.5. Penny-shaped Dugdale-Barenblatt crack.

32. Dugdale-Barenblatt cracks 293
tension S is applied at infinity, the tip of the crack is subjected to an infinitely
large stress even for a small S. The stress, however, is relaxed if a plastic zone
at the neighborhood of the crack tip is developed, as indicated by the hatched
area с < r<a in Fig. 32.5. When the crack surface is normal to the tensile
direction and the maximum shear theory of yielding is employed for the
plastic domain, Dugdale's hypothesis requires the following boundary condi-
conditions for an axially symmetric problem:
o2 = S at 2 = oo,
°zr= агв ~ 0 at 2 = 0,
az = 0 at 2 = 0, 0 < r < c, C2.46)
\{oz — oe) = k0 at 2 = 0, c<r<a,
w = 0 at 2 = 0, a<r,
where w is the displacement in the 2-direction and k0 is a point on the
stress-strain curve as shown in Fig. 32.2. Since the material in the plastic zone
is work-hardening, the stress state in this zone is not uniform. However, for
mathematical simplicity we assume that the stress state is uniform in the
plastic zone. It is convenient to consider, instead of C2.46), the following
boundary conditions:
C2.47)
Proof that \{az — ae) gives the maximum stress in Tresca's yield condition will
be given after the solution has been obtained. Keer and Mura A966) have
solved this problem as described below.
According to Inglis A913), Collins A962), and England A963), the solution
for the axially symmetric problems can be expressed as
32ф Э3ф
3z2 3z3
3z2 3r2 r drdz'
3r 3z2
°z--
°zr
°z~-
H<
W =
= 0
= o,e = 0
= -5
(Jz — Off) ^ Kq —
= 0
1 о
at
at
at
at
at
Z = 00,
z = Q,
z = 0,
z = 0,
2 = 0,
0
с
a
<
<
<
r
r
r
<
<
c,
a,

294 Chap. 5 Cracks
Oze=iJ!±- C2.48)
r 30 3z2
Э2ф 2v Зф 3
O = H
— -^- +z-
Ьг2 г Эг Ъгг 3z
where
C2.49)
/@ = -/(-<)•
The unknown function /(/) is determined from the third and fourth condi-
conditions in C2.47), using C2.48). Since we have
2 = (r2-/2I/2 (r>t)
and
[(
the derivatives of <#> as z -* 0 become (when /@) = 0)
@<r<a)
I (t2-r2I/2
= 0 (a <r < oo),

32. Dugdale-Barenblatt cracks 295
Conditions C2.47), therefore, lead to
Г drJ0 (r2_,2)V2
.r tf< Л At \
-k— \S (c <r < a).
The function f(t) can be determined from C2.51) by the method described by
Green and Zerna A954),
= St @<t<c),
C2.52)
It is seen that \(az — ae) is the maximum shear stress.
Using C2.50) and C2.48), we have
oz - ад = 2k0 - S,
oz-or = 2k0-S-{2k0-(l + 2v)\S}-r, C2.53)
or-ae={2k0-(\+2v)\S}-r,
at z = 0, с < r< a. Furthermore, we have
by applying integration by parts to the second formula in C2.50). The
requirement that the stress singularity vanish at r = a leads to
Да) = 0 C2.55)
which becomes, from C2.52),
C2.56)

296 Chap. 5 Cracks
The displacement w can be easily obtained from C2.48) as
.-A
A-2,)
Ш
vl/2
The value at the tip of crack is
с2\1/2A, с2
С _! С
с с /-1 cos ^c/a?)
+ 2l
Equation C2.56) can be approximated as
and equation C2.58) as
24A -v)kQ 1л
2w
77A —
C2.58)
C2.59)
C2.60)
when c/a = 1 and S//c0 <c 1.
Eliminating A - c/a) from C2.59) and C2.60), we have
C2.61)

* 33. Stress intensity factors for an arbitrarily shaped plane crack 297
If it is postulated that the crack becomes unstable when the crack opening
displacement 2w reaches a critical value Ф, then the critical applied stress can
be expressed as
S={3(l-v)(l-2,)c
The ratio of C2.22) to C2.62) is about 0.42 when v=\ and the к for the plane
stress case is used. This means that the slit-like crack is 0.42 times weaker than
the penny-shaped crack for the same crack length с
It will be shown that the approach using the dislocation distribution and
that using the Dugdale plasticity hypothesis are equivalent for the case of the
penny-shaped crack.
According to Kroupa A960), the stress a2 due to a circular dislocation
located on the plane z = 0 is
where b is the Burgers vector in the z-direction and s is the radius of the
dislocation. Keer and Mura A966) have shown that the equivalent stress and
displacement fields can be obtained when the distribution of dislocations is
chosen as
C2-64)
where f(t) is defined by C2.52).
* 33. Stress intensity factors for an arbitrarily shaped plane crack
The three-dimensional elasticity problem of finding the stress distribution near
a flat crack embedded in an infinitely extended homogeneous, isotropic solid,
and opened up under prescnbed internal normal pressure, has been solved
analytically by a number of investigators for cracks having circular and
elliptical contours. The problem becomes much more difficult, and analytical
solutions are not available when the crack contour is other than circular or
elliptical. For such cases, attempts have been made to formulate the problem
in terms of integral equations which might be solved by numerical techniques.

298 Chap. 5 Cracks
An integral equation formulation of the general three-dimensional elasticity
problem has been proposed by Kupradze A953, 1965), Kinoshita and Mura
A956) and Mikhlin A965). This formulation is known as the boundary
integral equation (BIE) method. The method is derived from Green's formula
D8.21). One obtains the limiting form of Somigliana's identity for displace-
displacements inside the body due to known surface tractions and displacements,
when the point at which the displacement is evaluated approaches the surface
of the body. This results in a system of coupled integral equations for the
unknown surface tractions and displacements, as shown by equations D8.27)
and D8.28) for the elasto-dynamics. The idea is equally applied to the
elasto-statics (see Kitahara 1985).
Numerical approaches of the boundary integral equation method have been
developed by Rizzo A967), Cruse A969), Lacht and Watson A976), Banerjee
and Butterfield A981), and Brebbia, Futagami, and Tanaka A983), among
others. Cruse A973) has reported an application of the numerical method to
the solution of the penny-shaped crack problem. Bui A975, 1977) obtained a
singular integral equation for the crack displacement discontinuity. The shape
of the crack can be rather general and the domain of integration is the surface
of the crack. Bui has numerically solved the integral equation for the penny-
shaped, elliptical and square crack under constant normal pressure. Weaver
A977) has derived from Somigliana's equation a singular integral equation for
the unknown dislocations which are defined over the crack area only. He has
numerically solved the problem of a crack in the form of a rectangle under
constant normal pressure.
Considerable work in this area has been done by Soviet researchers. One
formulation is given by Panasyuk A968), who has reduced the crack problem
to the solution of a singular integral equation for the crack displacement, but
has not elaborated on the development of a solution technique for an
arbitrarily shaped crack. He also discusses in some detail an approximate
solution for an almost circular crack. Martynenko A970), using Green's
functions of a Riemannian space, gives the exact solution for a circular crack.
He also obtains an approximate solution for a crack in the form of a strip
which becomes exact when the strip becomes a half-plane. Andreykiv and
Stadnik A974) consider a plane crack whose contour is a convex, closed
piece-wise smooth line, consisting of n (n = 1, 2, ...) circular arcs forming
sections of the circumferences of circles, each of which encloses the entire
crack domain, and m (m = l,2, ...) straight-line segments. They reduce the
problem to the solution of a system of two-dimensional Fredholm integral
equations with the stresses unknown outside the crack domain. They numeri-
numerically solve this system for semi-circular and square cracks under constant
pressure.

* 33. Stress intensity factors for an arbitrarily shaped plane crack
299
Fig. 33.1. An arbitrary plane crack S on x3 = 0 plane.
What follows is a formulation of the plane crack problem in the work of
Mastrojannis, Keer, and Mura A979). To test the formulation, a numerical
analysis is given involving plane cracks of various contour shapes embedded in
an infinite isotropic solid and subjected to uniform and linearly varying
internal normal pressure.
Consider an unbounded homogeneous elastic solid weakened by a plane
crack S having a smooth contour dS. Let x, (/ = 1, 2, 3) be the Cartesian
coordinates of the point jc, and let the plane x3 = 0 be the plane of the crack
(Fig. 33.1). The faces of the crack are loaded symmetrically with respect to the
x3 = 0 plane with an arbitrary normal pressure of intensity Р(хъ х2)', at
infinity, the solid is stress-free. It is necessary to determine the stress
o33(xx, x2, x3 = 0) in the region 5° outside the crack domain, the normal
displacement u3(xu х2, x3 = 0) of points inside the crack, and the stress
intensity factor kx (Mode I) along the crack edge.
The determination of the stress оъз(хъ x2,0) and the displacement
M3(*j, x2, 0) reduces to a problem for the elastic half-space x3 > 0 with the
following boundary conditions (Sneddon 1966):
и3(*1, jc2,0) = 0 for(x1; *2)
o33(x1,x2,0)=-P(x1,x2) tor(xltx2)
o31(Xl, x2,0)=a32(x1, *2,0) = 0 for(xlf x2)
The equations of equilibrium
+0 ( 123)
C3.1)
C3.2)

300 Chap 5 Cracks
and the conditions of zero tangential stresses on the plane x3 = 0 are satisfied
by taking the displacement vector (e.g., Panasyuk, 1968) in the form
where v is Poisson's ratio and Фг, Ф2, and Ф3 are harmonic functions of x
such that
The stresses a33, a31, and a32 can be expressed in terms of Ф3 as follows:
() ЛФ
where ? is Young's modulus. Thus, the elastic problem posed by equations
C3.1) reduces to that of finding a harmonic function Ф3(х) satisfying the
mixed boundary conditions
Ф3(хг, х2,0) = 0
дФ3(хи х2, х3) "" ~2'
г, x2)
C3.6)
To solve this problem, the harmonic function Ф3(дг) is expressed as a
Fourier integral (Andreykiv and Stadnik 1974),
C3.7)

* 33. Stress intensity factors for an arbitrarily shaped plane crack
where Л(?15 ?2) is an unknown function. Letting
2A -,2)
301
g(xlt x2) =
2-nE
2A-
x^, x2)
x, x2,Q)
C3.8)
2-пЕ
we obtain from C3.6) and C3.7)
iSi + jc^MUi. €2) d€i d?2 C3.9)
which, upon application of the inverse Fourier transform theorem, gives
1, tj2) diji dij2- C3.10)
Using C3.10) in C3.7) and noting that
2-7T
и
1/2 '
C3.11)
we have
x2,0)=
Htu Z2) <Hi
/I.
\(Y -t \2+(x
\\л\ i\) "f \Л2
°33Uu Si,
1/2 Г
C3.12)
Finally, from the first condition of C3.1), one obtains the following integral
equation for the stress а33(л;1, х2, 0) in $°:
If,
1/2
for(x1;
C3.13)

302 Chap. 5 Cracks
Once the stress о33(хг, x2, 0) acting on the plane of the crack is known, the
stress intensity factor kY (Mode I) for a plane crack can be determined from
the equation
*,= lim Bт7г1I/2а33(/1, Г2 = О, *3 = 0), C3.14)
r0
where tx, t2, and t3 constitute a local orthogonal coordinate system on the
crack contour, as shown in Fig. 33.1.
The analytical solution of C3.13) for an arbitrarily shaped crack appears to
be beyond the present mathematical techniques. For this reason, numerical
methods must be used in order to obtain specific results. It is well known that
the stress ct33(?i, ?2, 0) vanishes at infinity and has a square-root singularity
on the smooth contour 3S of the crack. Therefore, we assume that the stress
distribution а33(?г, ?2, 0) m C3.13) has the form
а33(|1,|2,0) = Б(|1, ^2)/[2^(|„ ?2)]V2 fot(SltS2)eS0, C3.15)
where 5(?i> ?2) *s an unknown bounded function, and the function w(?1; ?2)
represents the shortest distance from the point (?l5 ?2) to the crack contour.
Thus, н'(^1, ?2) behaves like tx in C3.14) near the crack neighborhood, and, at
a point at infinity, w(|1( ?2) -> 00. Rvachev A975) and Goncharyuk A972)
have given a method for the construction of и>(?ъ ?2).
Let the smooth crack contour dS Ъе approximated by a set of n straight
line segments with end-points Mi(xv уг), Nj(x2, y2) (' = 1, 2, 3,..., n), where
xx, Vj are the coordinates of point Mf. When the crack contour is convex, the
shortest distance of point (?2, |2) from the /th straight line can be written as
/•(€1, ?2) = d, - €x cos et - i2 sin 0,, C3.16)
where d, is the normal distance from the ongin of the |1; ^-coordinate system
to the straight line /)(|1( C2) = 0, and (cos 6t, sin 6,-) is a unit vector along d,.
Then, we write
w(€i. «2) = |//(€i, ЫЛгШи t2)---Aifn(ui> €2)|, C3-17)
where
C3.18)

* 33. Stress intensity factors for an arbitrarily shaped plane crack 303
The right-hand side in C3.17) has been called the R,-conjunction by Rvachev
A975). It defines analytically the minimum value among |Л(?1, $2) |.
|Л(^1> ?г)!>••¦> |/n(?i> ?2) I- Furthermore, we assume that the function
|], ?2) has the following form:
JJs
Ui-4i) +(€2-42)
forU,, «2)e5°, C3.19)
where G(|t, ^2) is a new unknown function to be determined.
With the above assumption, C3.15) becomes
it t ^ G(Si. ?2)
e({€0)"
for^j, |2)g5°, C3.20)
and C3.14) leads to
, €2) JJ
s U
C3.21)
The integral equation C3.13) is reduced to the following one which defines
(?(?„ €2) in S°:
AT
i. €2)
ls x2)]
l/2

304 Chap. 5 Cracks
X
1/2
1/2
for(x1( ^2)eJ°, C3.22)
where AT is a small region of the domain S°, inside which point (xb x2) is
located, and
C3.23)
The sum of the third and fourth terms on the right-hand side in C3.22) are
zero, but these terms are artificially added to avoid numerical integration at
the singularity ? = x. The fourth integral is analytically performed as shown
below.
When AT is taken as a quadrilateral with the centroidal {хъ х2), we have
= Дх log(C1) + R2 \og(C2) + R3 log(C3) + R4 log(Q), C3.24)
where
C3.25)

* 33. Stress intensity factors for an arbitrarily shaped plane crack 305
and where (X,, УД (Х2, Y2),(X3, У3), and (X4, Y4) are the coordinates of the
corner points of the quadrilateral. The expressions for R2, C2, etc., can he
obtained from C3.25) by the cyclic permutation of A, 2, 3).
To treat the integral equation C3.22) numerically, the infinite domain of
integration, 5°, is replaced by a finite domain, Sf, enclosed between the crack
contour 35 and a similarly oriented and shaped contour 35j? at some distance
from 35. The finite domain 5^ is then divided into N quadrilaterals over each
of which the value of the unknown function G(?1; ?2) is assumed to be
constant, where (?l5 |2) defines the centroid of the corresponding quadri-
quadrilateral. Also, the crack domain 5 is divided into M triangles and M quadri-
quadrilaterals, where M is equal to the number of the line segments that compose the
crack contour. Replacing the integrations over 5 and Sp by the sum of
integrals over quadrilaterals and triangles, the solution of C3.22) reduces to
the solution of a system of N algebraic equations with ./V unknowns (the
values of G(?lt ?2) on each quadrilateral in 5^).
* Numerical examples
The integral equation C3.22) has been solved and the stress intensity factor кг
calculated for cracks subjected to uniform and linearly varying internal
pressure, i.e.,
P(xu x2) =P0, ,
, г 2' ° C3.26)
P(xlt x2)=P0 + PlXl,
where Po and P1 are given constants. The cracks have contours in the shape of
a circle, an ellipse, an egg, a square, and a triangle. The contour of each crack
is approximated by 48 straight line segments, and the corners of the square
and triangular cracks are rounded. The outer contour dS° is taken at a
distance from the crack edge 3.5 times the radius of the inscribed circle.
Choosing the center and the orientation of the coordinate system properly and
utilizing the resulting symmetry about the ^-axis, we reduce the solution of
integral equation C3.22) to the solution of a system of algebraic equations
with 144 unknowns. The running time for the solution of each crack problem
with the two loading cases was 143 seconds on the CDC 6600 Computer, the
only input data being the coordinates of the 48 points on the crack contour,
the values of Po and Px, the normal distance between 35 and 35°, and the
number of unknowns.
Figures 33.2 and 33.3 are plots of А:1/2/>0(Л/тгI/2 against the polar angle
в, where R is the radius of the inscribed circle. The ratio P\/Po is taken to be
0.2. The results for the square crack under constant normal pressure seem to

Chap. 5 Cracks
1.420
1.310
I""
gf Ю90
.980
.870
7fifl
\ l R/
;;
A
P, /P0=0.2~/ \
40 80 120 160
ANGLE 9 (DEGREES)
Fig. 33.2. The stress intensity factor along an egg-shaped plane crack.
1.280
1.120 ¦
.800
.480
40 80 120 160
ANGLE в (DEGREES)
Fig. 33.3. The stress intensity factor along a rectangular plane crack.

34. Crack growth 307
be identical (excluding corners) to the results obtained numerically by Weaver
A977).
For polygonal plane-cracks, the stress singularity at points of abrupt
changes in the slope of the crack contour is less severe than the square-root
singularity along the smooth portions of the contour. For this reason, the
stress intensity factor, as defined by B9.21), at those points assumes a zero
value.
The method discussed here differs from those employed by Cruse A969,
1973), Bui A975, 1977), Weaver A977), Andreykiv and Stadnik A974), and
Hayashi and Abe A980). The present method results in a single integral
equation for the function G(?t, ?2) characterizing the normal stresses outside
the crack domain.
Although the support of the integral equation C3.22) is infinite and
truncation of the domain of integration introduces some error, the numerical
results obtained by the present method show only 0.5 per cent error for the
circular crack and 1.85 per cent error for the elliptical crack compared with
the known analytical results. This is extremely satisfactory compared with the
results reported by Cruse A973), 16 per cent in error, and by Bui A977), 20
per cent in error, for the circular crack under constant pressure. In Bui's
method, the support of the integral equation is finite (integration over the
crack domain only), but the treatment of the singularity seems to be the main
difficulty. The satisfactory results of the method offered here are attributed
mainly to the choice of the stress distribution a33(?i, ?2> 0) in the form shown
in C3.20) with the unknown function G(?l5 ?2)-
34. Crack growth
A crack tends to grow when an applied stress reaches a critical value. The
crack propagation may be dynamic for brittle materials. If the material is
ductile, the propagation may be quasi-static and the crack length may increase
slowly with a load, maintaining at each instant a state of equilibrium. In the
case of fatigue, cracks usually grow in the static fashion for any cyclic loading
unless the tips of the cracks are extremely slanted or residual stresses are built
up to oppose the motion of the cracks.
Energy release rate
Let us consider the static growth of a crack A in a body D subjected to the
applied force Ft on its surface S. The displacement and stress fields are
denoted by м, and atj, when the crack surface is A. We are interested in
evaluating the change of energy during crack growth 8A (Fig. 34.1) in order to

308
Chap. 5 Cracks
CRACK
Fig. 34.1. Crack A grows by SA.
define the generalized force at the crack tip necessary for the crack extension.
Changes of the displacement and stress fields due to the crack extension 8A
are denoted by 8ut and 8atj, yielding
dS
= jfo,
«
,-,,
dS,
C4.1)
where В is a surface of discontinuity of 8uh and и is the normal vector on
B+ = B. For most cases, В is 8A. For the BCS model, В includes the plastic
zone and [5м, = 8м, (on B+) - 8ut (on B~). Equation C4.1) has been derived
from js FiSui dS = Js а^п^иг dS by applying Gauss' theorem, where the
boundaries of D are S, A+, A', B+, and B~ (see Fig. 34.1) and where
oij} = 0 in D and a^rij = 0 on A + = A. The strain |(",>y + wy>,) is the sum of
elastic strain etj and plastic strain ef^,
C4.2)
Therefore, C4.1) can be written
JFfiu, dS - f aiJ8eiJ dD = f a^fj dD - f аи^[8щ] dS. C4.3)
JS JD JD JB

34. Crack growth 309
fs FjSuj dS is the work done by applied force Ft during the change of crack
surface 8 A and can be written as 8 fs Fiui dS under the assumption that F
does not change during the infinitesimal crack growth. /o o^Se^ dD =
8[\jDaijeiJ d-D] is the change of elastic strain energy. /д <r,7Se? dD is the
plastic work done during the crack growth. Finally, — fB о^иДЗи,-] dS is a
positive work done at the crack tip during the crack opening [8ut].
The left-hand side in C4.3) is identical to -S(AW) when e? = 0, where AW
has been defined by B5.14).
The expression
П = hjD%eu dD - JFiUi dS C4.4)
J
is the mechanical energy portion of the Gibbs free energy. The Griffith
fracture criterion can be written as
8в = 8(П+2Ау) = 0, C4.5)
where G is the Gibbs free energy and у is the surface energy per unit area of
the crack surface. Since the right-hand side in C4.3) expresses irreversible
energies, — 8П can be interpreted as the energy release during the crack
growth 8A. If a slit-like crack is changed by 8A, the energy release rate, Й7,-, is
defined from
9,81 = -Ш = / ч№ dD - [a^jlSu,] dS, C4.6)
where S?, is the displacement (change of position) of the crack tip due to the
crack extension 8A. For a three-dimensional crack, the left-hand side in C4.6)
must be modified (see e.g. Palaniswamy and Knauss 1978).
For a slit-like crack, the generalized force acting on the tip of the crack is S7,-
(crack extension force). For the slowly growing plastic crack, the second term
in C4.6) can be neglected since it is of order (8AJ. Then, C4.6) means that
the plastic work during the crack growth 5?г = 8A is proportional to 8A. This
result agrees with the experiment reported by Lee and Liebowitz A978).
Irwin A958) has calculated — fSA а,-7-пД8м,-] dS for a slit-like elastic crack
with length 2 c under tension of2 in an elastic medium. According to the
elasticity analysis, а22 = к1/{2'тт(х1 — X)}1/2 and u2 = A - p)kl{2(X —
1/2V^:; where X is the xx-coordinate of the crack tip and kl = а2°2G7сI/2.

310 Chap. 5 Cracks
Irwin assumes that [6м,] in C4.6) is half of [8u2] = 2м2 = 2A - v)kl{2(X +
12^
= -^-*I**1- C4-7)
The crack extension force is obtained as
C4.7.1)
The crack extension per unit length creates the surface energy 2y. When 2y is
expressed in terms of кг and 2ju from B7.20) with C0.11), it becomes the
crack extension force C4.7.1). The same expression can be obtained from
C0.15) by dividing by 2 since C0.15) is the energy increase caused by the
crack extensions at the two tips of the crack. The same integral - fSA а,упД8ы,]
dS can be calculated for the Dugdale crack. From C2.20), [S«2]«(l +
K)-7rS2S^l/Sfik(j and ст22 = fcoEc = 5?г). Then, the integral becomes A +
(сOг52E|1J/8]ч, which becomes zero at the limit 8?x ->0. Yokobori and his
co-authors A966, 1968) and Rice A966) pointed out this paradoxical result.
This inconsistency with the Orowan-Irwin theory was also discussed later by
Kfouri and Rice A977). Wnuk A972, 1974) has postulated that the integral
domain in the second term in C4.6) is a material constant (process-zone size)
and has derived an expression for a quasi-static extension of a tensile crack.
Denoting the process zone by A, he has defined the crack separation energy by
dx1; C4.7.2)
where k0 is defined in C2.1).
From C4.13.1),
cab ь II,. lHv
= -2ko{u2(c + A, c)-u2(c, c)}
C4.7.3)

34. Crack growth 311
For small-scale yielding, C2.18) can be approximated as
-U-illogf}, C4.7.4)
where а — с = R and x1 — с = |. Then, C4.7.3) yields
&_ 8(l-v2)k%A fdR 1 1 4R\
irE \ dc 2 2 Л I
This is Wnuk's equation A972) equation. A similar result is obtained by Rice
and Sorensen A976), Kfouri and Rice A977), and Kfouri A979).
From C2.20) we have, for small-scale yielding,
C4.7.6)
where Q is a loading parameter. If Л and &f are constant, from C4.7.5) and
C4.7.6) we obtain a relation, Q = Q(c), which represents a stable crack
growth. The catastrophic propagation is determined by the condition dQ/dc
= 0. This idea has been developed by Cherepanov A968), McClintock A958),
Rice A968), and Wnuk A979), among others.
The J'-integral
The generalized force acting at the tip of a crack can be expressed in terms of
a path-independent integral (/-integral) of a Maxwell tensor of elasticity taken
over a surface. This fact has been developed by Eshelby A951) and later by
Rice A968) and Eshelby A970).
The three-dimensional expression of the /-integral is defined by
/,= (Wn,dS- /'o///i.i/|.,dS= (РцП/dS C4.8)
Jr Jr ' Jr
with
P,J=W8IJ-ouuiJ, C4.8.1)
where Г is an arbitrary closed surface (a closed curve for the two-dimensional

312 Chap. 5 Cracks
case) including the crack tip, и is the outward unit normal on dS, P^ is the
energy-momentum tensor, and W is defined by
ИО = ",/,,,„ C4.9)
where etJ¦= \{utj + "/,<)• It IS easY to show that J, is a path-independent
integral. If another path (surface) Г* is taken, application of Gauss' theorem
leads to /r_r. Wn, dS - jr-r*aijnjui,i dS = 0.
For nonlinear elastic materials, C4.9) can be integrated as
W--
f\ deu. C4.9.1)
If the plastic unloading is involved, we cannot have C4.9.1) from C4.9).
Denoting by V the domain bounded by Г, A+, B+, A", and B~ (Fig.
34.1), we write C4.8), by Gauss' theorem, as
j dD - jjijUijj dД + f+B°,jnju<J dS> {ЪА.9.2)
f+B
where W, = а{]иг jh aihj = 0, and oijn] = 0 on A. It has been assumed in the
derivation of C4.9.2) that Won A + + B+ is equal to Won A~ + B~. The first
and second integrals in C4.9.2) cancel each other and а,уиу on B+ is —o^rtj
on B~. Then, C4.9.2) becomes
C4.10)
where [u,,] = {utl on B+ = B) - (uu on B~).
On the crack surface, м, is multiple-valued. After the crack tip has moved
by 8?, the change of displacement can be written as
8м,. = и,.(х - SO - и,.(х) + Ащ, C4.11)
where Ли, is a single-valued function. Equation C4.11) is Eshelby's basic
assumption. Taylor's expansion of ut{x — 8$) about point x is
и,.(х-8?) =«,¦(*)-И/./8&- C4.12)
Then, C4.11) becomes
««<,¦=-и^ + Ди, C4.13)

34. Crack growth 313
or
[««/]«-K/l8*/> C4.13.1)
since AUj is a single-valued function. Therefore, we can write C4.10) in the
form
/,«*,=-jTo,,/iy[8M,]dS. C4.14)
From C4.6) and C4.14) we have, for the slit-like crack,
f
JD
4D + Wi- C4.15)
When S?; = S|j = 8A and Sefj = 0, we have
*!=/,. C4.16)
The meaning of the /-integral in plane problems and in the absence of plastic
deformations, is clearly shown by C4.16); however, the same is not the case
when plastic deformations are involved. Miyamoto and Kageyama A978)
modified the /-integral for plastic deformations. It should be noted that /,
becomes zero when the material is perfectly plastic and [8м,-] is of the order of
8|1( since В = 8%! and ajJnJ = k0 in C4.14). J1 becomes ko8, for the BCS
model, where 5, is the crack opening displacement at the crack tip.
If dislocations in the neighborhood of the crack tip are displaced by the
same amount as the displacement of the crack tip, we can write
8tf,=-eJlh8iiahi, C4.17)
where ahj is the dislocation density tensor defined by G.33) [see C8.18)].
Therefore, Mura A979) has written C4.15) as
9, = - jatjtjlhahl dD + /,. C4.18)
The integrand in the integral represents the Peach-Koehler A950) force on
dislocations.
Other path-independent integrals related to energy release rates have been
investigated by Knowles and Sternberg A972) and Budiansky and Rice A973).
The /-integral is also used for investigating singular behavior at the end of a

314 Chap. 5 Cracks
tensile crack in a hardening material (Hutchinson 1968, Rice and Rosengren
1968).
Eshelby A970) has suggested a broader application of the energy-momen-
energy-momentum tensor to phase transformation problems. A martensitic transformation
region В undergoes a change of natural shape and size, that is, a change of
inhomogeneous inclusion. He writes the change of the mechanical energy
C4.4), caused by a migration 5?, of the inclusion surface S, as
C4.18.1)
where [PtJ] = P/y(out) - Pu(m), as defined in F.4). Since
C4-18-2)
where Э/Эи denotes differentiation along the normal, C4.18.1) is equivalent to
the statement that there is an effective normal force
C4.18.3)
per unit area of the interface. Equation C4.18.3) with F,= 0 can be used to
find the equilibrium position of phase and twin boundaries in the presence of
stresses produced by the transformation itself, or applied externally, or both.
Fatigue
It is well known that materials fail under repeated (cyclic) loading and
unloading at stresses smaller than the static fracture stresses. The magnitude
of the stress required to produce failure decreases as the number of stress
cycles increases. Micro-cracks are created in early cycles and they grow with a
velocity accordmg to a certain law; e.g. the Paris law A964). Weertman A966,
1973) has derived the Paris law by using the BCS model. A similar extension
of the BCS model to fatigue is given by Lardner A968), Bilby and Heald
A968), McCartney and Gale A971), and Mura and C.T. Lin A974). Recently,
Budiansky and Hutchinson A978) have used the Dugdale method for the
analysis of fatigue crack closure. Tanaka and Nakai A984) extended the
analysis to a finite crack.
Let us consider a sht-like crack (| x11 < c, x2 = 0) as shown in Fig. 32.1,
which may stand for an initial crack or a sharp notch. When a22 = S is first
attained at infinity, a plastic zone develops in the domain (с < \хг\ < a, x2 =

34. Crack growth
315
0). The displacement is given by C2.20), where k0 is the flow stress at this
stage. The plastic zone is assumed to be small compared with c, and kQ can be
defined uniformly throughout the plastic zone. If the first unloading is -AS
(the applied stress is reduced to S - AS), the change of stress and displace-
displacement is defined by the boundary conditions similar to C2.1),
О 0 = 1,2)
-AS
О, а91=0
atx2 =
I 1 1 ^ >
с< |^| <alt
at x2 = оо,
at л:2 = 0, а < |хх |,
C4.19)
where с < | хг \ < ax is a new plastic zone caused by the unloading. The stress
and displacement in C4.19) are increments of the state descnbed by C2.1).
The real stress and displacement at the unloaded state are the sum of those
corresponding to C2.1) and those corresponding to C4.19). The flow stress in
the new plastic domain is -(k0 + Akx), under the assumption that there is no
Bauschinger effect (see Fig. 34.2). The elasticity solution for the boundary
conditions C4.19) is similar to C2.20). It is obtained from C2.20) by changing
S to -AS and also changing &0 to -2k0- Akx. It hence follows that
277JU.
сA-с/в1).
C4.20)
The quantities corresponding to the subsequent loading by AS (the applied
Fig. 34.2. Work-hardening stress-strain curve.

316 Chap. 5 Cracks
stress becomes S again) are obtained in a similar way: we use AS and
k0 + Akx + k0 + Ak2 in place of -AS and -2ko-Ak1 in C4.20). Then,
1 - c/a2 = W{AS/Bk0 + Akx + Ak2)}2,
. + Ak,) C4.21)
<\-с/аг).
After n cycles of loading and unloading, we have
1 - c/an= U2{AS/{2k0 + Akn_, + Akn)}2,
п) C4.22)
c(l-c/an).
It is noted that
a > аг > a2 > ¦¦¦ > an C4.23)
for work-hardening materials, and
a > аг = a2 = ¦¦¦ =an C4.24)
for perfectly plastic materials.
Suppose that the initial slit starts to grow after the critical number of cycles,
N. We seek to establish some relation between the applied stress, the slit
length, and N. For this purpose we eliminate A — c/an) from the two
equations in C4.22) to obtain
, + Akn_x + Akn)
C4.25)
We assume that the crack starts to grow when the total plastic work done at
the tip of the crack attains a critical value, U. The total plastic work is
2kou2+T,^12(ko + AknJu2, where u2 in the first term is C2.20) and the
second u2 is C4.25), and the factor 2 comes from the work done at the upper
and lower surfaces at the crack tip. The other factor 2 in the second term

34. Crack growth 317
comes from the fact that N refers to both N loadings and N unloadings.
Then, our energy criterion leads to
C4.26)
Sjtl oJU.
or
C4-27)
where AK = (тгсI/2 AS and K={mcf/2S. К is called the stress intensity
factor and А К is called the stress amplitude intensity factor. If the slit (length
2c) is located at the center of a strip (width 2L), АК=(чтсI/2 AS[{2L/ttc)
tanG7c/2L)]1/2. This AK is also used approximately for a double edge-notched
strip (notch length a and strip width 2L). For a single edge-notched strip
(notch length с and strip width 2L), the correction factor, to be multiplied to
(ttcI/2AS, varies from 1.15 (for c/L = 0.1) to 2.86 (for c/L = 1).
From the first equation in C4.22), we have, approximately,
C4.28)
The rate of crack growth is
dc/dN=(aN-c)/N C4.29)
which leads to
Ac чгA + к) , . „-,* II„ 1 + к„,\ C4.30)
where kN = ko + AkN. This is the fourth power law of Paris A964). It has been
assumed in the derivation that the crack grows in the same fashion after N
cycles where the initial crack in the second step is the final crack in the first
step. Experiments (e.g., Izumi and Fine, 1979) seem to support the tendency
shown in C4.30).

318 Chap. 5 Cracks
A similar result can be obtained for the anti-plane problem where a shear
stress a°2 = 5 is applied at infinity and a°2 is cyclically changed with ampli-
amplitude AS. Then, we have
V '
If a penny-shaped crack is considered under cyclic tension with amplitude
AS, we have, from C2.59) and C2.60),
aN - с = A - 2vfc(ASJ/32(k0 + AkNf,
w = 3A - v)(l - 2r)c(ASJ/4vli(k0 + Ак„),
assuming Akn « Akn_x. Our crack growth assumption leads to
N
U=2k0w+ ? 2(ko + AknJw, C4.33)
where w in the first term is C2.60) and w in the second term is given by
C4.32). Then,
U= 3A - v)(l - 2v){cS2 + Nc(ASJ)/nfi, C4.34)
and, therefore, C4.29) gives
i?- = Ъ{\-у){\-2у)\{АК)"
where kN = kQ + AkN, K=2(c/-nf/2S, and AK= 2(c/irI/2AS.
Comparing C4.31) and C4.35), it can be said that the crack growth rate of
a slit-like crack is about 100 times that of a penny-shaped crack.
Weertman A981) has derived a second-power, Paris fatigue crack growth
equation for intrinsically ductile materials. Through repetition of forward and
reversed crack tip shear sliding, a crack tip can actually advance an increment
each stress cycle, as observed by Laird and Smith A962) and Neuman A974).
Weertman's theory is supported by experiments done by Kobayashi and Mura
A983).

34. Crack growth 319
Dynamic crack growth
When a crack extends at speeds high enough to necessitate the inclusion of
inertia effects, the associated kinetic energy must enter the energy argument of
the previous subsections. In recent years a rather extensive body of literature
has become available for the analysis of dynamic problems of crack propa-
propagation in homogeneous, isotropic, linearly elastic solids. Erdogan A968),
Achenbach A974), Freund A975) and Kanninen A978) have presented review
articles on this subject.
Let us consider the dynamic growth of a crack A in a body D subjected to
an applied force Ft on the surface S of D. The rate of work done by Ft is
AS
= / atJutJ AD+ f puiui AD- f оип^щ] AS, C4.36)
Jd jd jb
where В is the domain where ii, has a multiple value. The right-hand side in
C4.36) has been derived from js a^rifti AS by applying Gauss' theorem,
where аиj = put in D, о{]п} = 0 on А, [и,-] = м, (on B+) — м, (on B~), and л
is the unit normal on B+ = B. Since i(«;,y + w, ,) = e,7 + efy, C4.36) can be
written as
(Ftut AS - (d/dt)U f °,jeu AD+\( рй,щ
•>S \_ JD JD
f
AD - [о,л[й,] AS = 9& C4.37)
for a slit-like crack, where ei} and efj are elastic and plastic strains, respec-
respectively, ^, is the energy release rate, and ?, is the velocity of the crack.
Achenbach A974) further investigated ^, for a slit-like crack in an elastic
medium as Irwin A958) did for a static crack [see C4.7)]. Suppose at time
t = tQ a slit-like crack of width 1c starts to grow and at time t > t0 the position
of the crack tip is xl = X(t - t0) (see Fig. 34.3), where X(Q) = 0. The singular
parts of the stresses in the plane of the crack are of the general form
о2Ахх,0, t) = T2i[Xl-X(t-t0)}-1/2, C4.38)
and the singular parts of the particle velocities assume the forms

320
Chap. 5 Cracks
Fig. 34.3. Crack growth.
The functions T2i and Ц, which generally depend on time and on the
geometrical and material parameters, are termed the stress intensity function
and the velocity intensity function, respectively. Achenbach has evaluated
rX(t-to) + ( .
'x(t-tn)
C4.40)
where e is an arbitrarily small positive number. More specific expressions for
T2j and Ц were obtained by Achenbach A974) for the case of anti-plane
shear. The case of plane strain is examined by Nuismer and Achenbach A972)
and Freund A972). For the symmetric crack opening (Mode I), the instanta-
instantaneous hoop stress near the crack tip may be expressed as
v)Te\6, v),
C4.40.1)
where 7^@, v) = 1, k^t, v) is the Mode I elastodynamic stress intensity
factor, and v = ?: is the uniform velocity of the crack. The function Tj(8, v) is
universal because it is independent of the overall geometry and the loading.
The conditions governing crack motion can be expressed in terms of
kY(t, v) and an experimentally determined critical value that is assumed to be
a material property. According to a brief review by Achenbach and Kanninen
A978), the initiation of crack growth is determined by
kl(t,0)=Kd(d),
C4.40.2)
where 6 represents the loading rate. For perfectly brittle fracture, Kc = Kd.
Similarly, for a propagating crack,
kl(t,v) = KD(v), C4.40.3)

34. Crack growth 321
where KD is known as the dynamic fracture toughness. A crack is arrested
after time ta when kx < KD for all t > ta.
The /-integral C4.8) can also be modified for the dynamic case, where we
add the kinetic energy density К to obtain
/,= f(W+K)n,dS- /'cWii/dS, C4.41)
where
K=\pu,ui C4.42)
and W is defined by C4.9). Multiplying C4.41) by ?, and applying Gauss'
theorem, we have, for a slit-like crack,
JAt = j (auui,ji + Puiui,Mi dD - j OjjUijJ, &D
- f puiUiJ, dD+ f o^njuj, dS, C4.43)
where V is interpreted as being bounded by Г, A+, B+, A~ and B~, and W
and К on A+ + B+ are assumed equal to W and К on A~ + B~. uu on B+ is
not equal to w,_, on B~ as seen in C4.13.1) or
[",]=-[«„,]!, C4.44)
by dividing C4.13.1) by dr. The first and third terms on the right-hand side in
C4.43) cancel each other. Furthermore, the second and fourth terms also
cancel each other if V is sufficiently small, since from C4.13)
u'--«'-f C4.45)
Therefore, C4.43) becomes
njlu^dS. C4.46)
Then, the flux of energy release C4.37) can be written as
9,i= fo.jifjdD + J^. C4.47)
J r>

322 Chap. 5 Cracks
For the elastic crack of Mode I, we have
^ = Л=Ц^(<^12, C4.47.1)
where E and v are Young's modulus and Poisson's ratio, respectively, and A
is a geometry-independent function of the crack speed given by
- B - *
Here cL and cT are the longitudinal and shear wave speeds, respectively. The
function A(v) is unity at zero crack-tip speed, and it increases monotonically
to become unbounded as v -»cR, where cR is the speed of Rayleigh waves in
the material. As v -* cR the elastodynamic stress intensity factor vanishes, and
we find ^j -> 0 as v-* cR.
Contrary to the static case, the /-integral C4.41) is path-dependent. How-
However, it can be path-independent if Г is chosen as a small loop around the
crack tip, or a rectangle made up of segments \xt — X(t - t0) | = e and
| x2 \ = S. An alternative expression for the energy flux,
Mi = f(W+ K)n,i AS + (oijtijui dS, C4.48)
Jг Jr
can be derived directly from C4.41) and C4.45). The above expression has
been obtained by Atkinson and Eshelby A968) and Freund A972).
Several solutions of the equations of linear elasticity have been found for
moving cracks. Yoffe A951) has treated a crack of constant length being
opened at one end and closed at the other end with a constant speed. Craggs
A960) has considered a semi-infinite crack extending at constant speed.
Broberg A960,1967) has treated crack growth, starting from zero crack length.
Baker A962), on the other hand, considered a semi-infinite crack which
suddenly appears in an elastic solid and then extends at constant speed under
the action of a spatially uniform and time-independent normal pressure on the
crack surfaces. In contrast to the work of Broberg and Baker, Freund A972)
has treated the extension of an existing equilibrium crack in a body subjected
to general time-independent loads.
When a Dugdale crack with a constant length is propagating with a

34. Crack growth 323
constant speed v, the crack-tip opening displacement has been calculated by
Kanninen A968) and Kanninen et al. A968). Their result can be written as
8{v)=A(v)8{0), C4.49)
where A(v) is the function given by C4.47.2) and 6@) denotes the value of the
crack opening displacement in the static case given by C2.20).
A more realistic solution has been obtained by Atkinson A967) by consid-
considering a crack expanding at a uniform speed with collinear strip yield zones.
The order of stress singularity at the elastic crack is 0.5, as seen in C0.10).
The order of stress singularity decreases when plastic deformation is allowed
at the crack tip. It decreases with the decreasing of the work-hardening rate, as
shown by Hutchinson A968) and Rice and Rosengren A968). It becomes zero
for the perfectly plastic solid. Recently, Achenbach and Kanninen A978)
showed that, for a moving elasto-plastic crack, the order of the singularity
decreases with increasing crack velocity.
Recent developments in experimentation, numerical methods, and elasto-
plastic dynamic fracture are reviewed in a special issue of Int. J. Fracture
edited by M.L. Williams and W.G. Knauss A985).

324 Chap. 6 Dislocations
Dislocations
Certain fundamental formulae for the elastic field caused by a dislocation loop
have been presented in Section 7. In this chapter, the integrations appearing in
these formulae are completed, and applications to the continuum plasticity are
discussed.
An extensive literature is available on solutions of elastic fields caused by
dislocations in various states of motion in bodies with various materials and
geometries. The books by Hirth and Lothe A982), Steeds A973), and Lardner
A974) contain explicit expressions for the elastic fields. On the physics of
dislocations, the following books are mentioned: Read A953), Cottrell A953),
Weertman and Weertman A964), Hull A965), Friedel A964), Suzuki A967),
and Nabarro A967). Reviews and proceedings on dislocation theory include
those by Seerger A955), deWit A960), Mura A968), Kroner A968), Simmons,
deWit and Bullough A970), Bacon, Barnett and Scattergood A978), and
Nabarro A979). Continuum theories of continuous distribution of dislocations
and plasticity are found in Kondo A955), Bilby A960), Kroner A958), and
Kunin A975).
35. Displacement fields
The displacement field of a dislocation loop L is expressed by G.6),
u^x)^ fcjlmnbmGiJtl(x- x')nndS(x') C5.1)
or, from C.19),
'), C5.2)
where 5 is the slip plane, L is the boundary of S, and n is the unit vector
normal to S.

35. Displacement fields 325
Parallel dislocations
Suppose that the slip plane S(x) is defined by
-oo<jc<oo, -R<y<0, z = 0. C5.3)
The plane S is a strip shown in Fig. 35.1. It should be noted that the parallel
dislocations (bounding S) are not necessarily parallel to the crystalline direc-
directions Xj, x2, x3. Vectors ? and x have the respective components (?l5 ?2, ?3)
and (jCj, x2, x3), in the x,-coordinate system, but have the components
(?, т), f) and (x, y, z) in the x, y, z-coordinate system that is,
y') + !;{z-z'). C5.4)
The unit vectors in the x, y, z-directions are denoted by v, m and n,
respectively.
The following calculations have been performed by Willis A970). Equation
C5.2) becomes
xexp[/{|(x - x') + v(y-y') + U}\ d? dr, df. C5.5)
Integration with respect to x' produces a factor 27г5(?), so that after integra-
integration with respect to ?, C5.5) becomes
y) + fz}]d4df. C5.6)
Next, we integrate with respect to f, regarding it as a complex variable. If
z > 0, we close the contour in the upper half of the complex f-plane (the lower
half-plane for z < 0) and then use Cauchy's theorem to obtain
C5.7)

326
Chap 6 Dislocations
«.С
Fig 35.1 Parallel dislocations bounding the slip plane S
where
and f ^ are the three roots, with positive imaginary parts, of
C5.8)
C5.9)
v, m, and л are orthogonal unit vectors shown in Fig. 35.1. Since
?iN,j(ii)/nk dD/d?k is a homogeneous algebraic equation of degree zero, we
divide the numerator and the denominator by tj5. Then C5.7) becomes, by
dividing the domain of integration,
X
exp{iij(.K -
dr,
-R
dyf[°
J —
, C5.10)

36. Stress fields
where fN are the three roots of
327
) = 0 C5.11)
with positive imaginary parts and lN denotes the complex conjugate of fN.
The integrals can be easily evaluated, since fN and fN are independent of tj,
and the domain of integrals is divided to have convergency. The result is
+
^—
log
C5.12)
A straight dislocation
The solution for a single dislocation along the x-axis is obtained by R -* oo.
The terms of log R are neglected, assuming that they are infinitely large
constants. Then, we have
N=\
. C5.13)
36. Stress fields
The elastic distortion of a dislocation loop L is expressed by G.16),
C6.1)

328 Chap. 6 Dislocations
dislocation segment L
Fig 36 1. Dislocation segment yz. Unit vector л is normal to the plane xyz
The stress field can be obtained from
Dislocation segments
C6.2)
Since an arbitrary dislocation loop can be approximated by the sum of a finite
number of dislocation segments, the solution for a dislocation segment is
fundamental for any shape of dislocation loop. The following calculation is
based upon the work of Willis A970), and Mura and Mori A976).
Let us consider a dislocation segment yz shown in Fig. 36.1. An arbitrary
point x' on the segment is expressed by
and the volume element in the ?-space is given by
C6.4)
where d5(|) is the surface element of the unit sphere S2 in the ?-space and
? = ?Д and | = | ? |. Willis has integrated C6.1) first with respect to / and
then with respect to ?, obtaining
C6.5)

36. Stress fields 329
-s'tf)
"I (y-x)=O
у-х
Fig. 36.2. Integral domain of the integral in C6.5).
In order to have a convergent integral, the exponential term in C6.1) has been
written as exp[|{-e + /f- (x - x')}], where e > 0. The limit e -> 0 has been
taken after the integration.
Njk(?) and .D(l) are homogeneous polynomials of degrees 4 and 6 respec-
respectively, and the integrand in C6.5) is an odd function of f. Thus, the integral on
the unit sphere S2 vanishes almost everywhere except on the two unit circles
satisfying ?-(y- x) = 0 and ?-(z-x) = 0; see Fig. 36.2. It is sufficient to
reduce the integral domain in C6.5) into the narrow strips of width 2e along
these two unit circles.
Let us introduce three unit vectors n, от*, and от, as seen in Fig. 36.1. и is
normal to the plane containing points x, y, z\ m* is orthogonal to я and
y — x; and от is orthogonal to n and z — x. The vector products от* х л and
m X n are directed parallel to the y — x and z ~ x directions, respectively.
The principal part of the integrand in C6.5) can be written as
1 1
When the integral on the narrow strip along the circle f-(j> —лг) = О is
considered, the first term on the right-hand side in C6.6) is used; on the other
hand, when the integral on the narrow strip along the circle f • (z — x) = 0 is
considered, the second term on the right-hand side in C6.6) is used. The plane
of circle ?'(y — x) = Q is perpendicular to (y — x) and, thereore, contains

330 Chap. 6 Dislocations
vectors я and от*. The plane of circle! • (z - x) = 0 contains vectors л and от.
On the narrow strip along the circle \-(y - x) = Q,
|= -от* sin ф + л cos ф + И(у-х)/\ y-x\, |A|<e<=cl, C6.7)
with parameters ф and /i, where h is infinitesimally small. Then the integral in
C6.5) defined on this narrow strip becomes
C27r
C2
Jq
Г (-m* sin ф +nt cos <j>)Nik(-m* sin ф +п cos <j>)
J-c D(-m* sin ф + п cos ф) sin ф{т* ¦ (z-y)}h\ у - x\ '
C6.8)
where
i-(z-y)= -sin ф{т*-(г-у)},
l-(y-x) = h\y-x\,
are used. Furthermore, we have
{m*-(z-y)} \y-x\=r\z-y\ =2Xareaof triangle xyz, C6 Ю)
(zh-yh)/\z-y\=vh,
from Fig. 36.1, and
f = '*, C6.10.1)
where r is the distance from л: to the dislocation line, and Cauchy's principal
value is taken for the last integral with respect to h.
A similar calculation is performed for the integral on the narrow strip along
the circle | • (г - x) = 0 by putting
|= -от sin ф + я cos ф + /г(г-лг)/|г-л:|, |A|<c«l. C6.11)

36. Stress fields 331
Then, C6.5) becomes
or
[1 (lit (~m* sin ф + nt cos <$>)Nik(-m* sin ф + n cos ф)
I
[1 (l
— — I
1 /-
IT I
2 /0
sin <J>D(-wt* sin ф + п cos ф)
w ( —W/ sin ф + ri/ cos ф)^( —iff sin ф + и cos
7чОф
sm фБ{-т sin ф + и cos ф) J
C6.12)
Bтг)
-т* sin ф +л cos ф)
x
2J0 [ 'd{-
т* sin ф + л cos ф)
cos ф / Nik( — m* sin ф + л cos ф) Njk(n)
D(-m* sin ф + л cos ф) ?>(")
m,Nik{-m sin ф + л cos ф)
D(—m sin ф + л cos ф)
Nik(-m sin ф + л cos ф) Nik(n)
D( — m sin ф + л cos ф) -D(«)
C6.13)
where /O27r(cos ф/sin ф)п[{Njk{n)/D(n)) &ф is added to the first integral in
C6.12) and subtracted from the second integral in order to eliminate the
singularities at ф = 0, m, and 2tt.

332 Chap. 6 Dislocations
By introducing the definition
%JNjk(-m* sin ф + п cos ф) cos ф
D(-m* sin ф + л cos ф) sin ф '
dф, C6.14)
(-m* мпф + л созф) ?(л
C6.13) can be written as
^{m*, n)-\Ilik{m, и)}. C6.15)
For an infinitely long straight dislocation, m* = —m, and m* and m are
perpendicular to v. Then, I/ik(m*, n)= -Ilik(m, n) and, therefore,
, П), C6.16)
where n = vXm.
Figure 36.3 shows the atomic arrangement around the center of a straight
edge dislocation in BCC Fe. This arrangement of atoms has been determined
OOOOOOOOoooo
oooooooo000o
ooooooo00000
O00OO°00O0O0
ooooo° oooooo
00000° OOOOOO
00000°° OOOo
oooo oooooooo
a [in]/2
Fig. 36.3. Atomic arrangement around the center of an edge dislocation in BCC Fe.

36. Stress fields 333
from C6.16) with C6.14) or C6.24) when v = [110]/Д, b = a[lll]/2, where a
is the lattice constant of BCC Fe. Looking through the BCC structure along
the [110] direction, we notice two types of A10) planes, alternatingly stacked
perpendicular to this direction. The atoms on these planes are denoted by the
filled and open circles. In this calculation, the elastic moduli are taken as
Cnn = 2.37 X 10n N/m2, C]]22 = 1.41 X 1011 N/m2, and C1212 = 1.16 X 10"
N/m2.
Willis' formula
There are several alternative expressions of C6.15) which lead to Willis' result
A970) and the result of Asaro et al. A973). Willis obtained the solution in
terms of residues of a complex plane. Employing the transformation
p = -cos ф/sin Ф, dp = йф/&т2ф, C6.17)
neglecting the terms which have been added and subtracted artificially, and
considering the fact that Njk(?)/D(?) is a homogeneous function of ? of
degree - 2, we write C6.14) as
(m*+Pnl)N'k(m*+Pn) л СХСЛ9\
D{m*+pn) d/, C6.18)
Ilik(m, n) is also transformed to a similar form. The integral in C6.18) can be
expressed in terms of residues in the complex p-plane by Cauchy's theorem.
Then, C6.15) becomes
3
N = l
C6.19)
where fN and f*N are complex numbers with positive imaginary parts and are
the roots of the following equations:
D(m + n{)=0,
, ч C6.20)

334 Chap. 6 Dislocations
Since D(m + n?N) is the sextic equation in fN, considerable computation is
required to find the roots of C6.20). Expression C6.13) is usually more
convenient for numerical calculation than C6.19).
The Asaro et al. formula
The result of Asaro, Hirth, Barnett, and Lothe A973) can also be obtained
from C6.15). The following notation is introduced:
ri = -m* sin ф + n cos ф,
m' = m* cos <j> + n sin ф,
{m>,n')s, = Csp,qm'pn'q, C6-21)
(ri,ri)~kl=Nik (»'
Then, we have
{ri, ri) = со82ф(и, п) — cos ф sin ф{(т*, п) + (n, m*)}
+ sirAf>(m*, m*), C6.22)
(m', ri) = со82ф{(т*, п) + (n, m*)}
-cos ф sin ф{(т*, m*) — (и, и)} - (n, m*),
where / is the unit matrix. By substituting the second equation into the first
and multiplying by (и, и) cos ф/sin ф, we have
{/со8ф/япф-(и, n)~\n, m*)-(n, n)~\m', n'))(ri, ri)~l
= («, n)~l cos ф/sin ф, C6.23)
where the third equation in C6.22) has been used. The second integrand in
C6.14) can be written as the ik component of n,{(ri, n')~l —
(n, л)} cos ф/sin Ф, which becomes from C6.23) «,{(«, n)~l(n, m*) +
(n, n)~l(m*, ri)}(ri, л'). Therefore, we have
~n,{n, nOX{{n, m*)st + (m', ri)s,}(ri, n%k'} dф. C6.24)

36. Stress fields 335
The expression C6.15) with C6.24) is the result obtained by Asaro et al A973).
These expressions can be represented in alternative ways, as given by Barnett
A972), Barnett and Swanger A971), Barnett et al. A972), Malen A970, 1971),
and Barnett and Lothe A973). These results illustrate further development of
the works of Eshelby, Read and Shockley A953), Stroh A958, 1962), and
Bullough and Bilby A954). On the other hand, Lothe A967), Brown A967),
and Indenbom and Orlov A968) have shown how the in-plane strain field of
the general planar dislocation loop is determined directly from the strain field
of a straight dislocation and the derivatives of these strains with respect to
variables describing the direction of the line in the plane as shown by G.17).
The details of the stress analysis of dislocations, along with the work of
Barnett et al. and Lothe and Brown, can be found in the review article by
Bacon, Barnett and Scattergood A978). The stress analysis of an infinite
straight dislocation based upon the work of Eshelby et al. and Stroh can be
found in the book by Steeds A973).
Dislocation loops
The elastic distortion of a dislocation loop L is calculated by the integration of
C6.15) along the dislocation line (see Fig. 36.4),
C6.25)
where the difference of Ilik at two neighboring points in C6.15) has been
expressed by the differential. The unit vectors m, n, and v are shown in Fig.
36.4. Vector и is normal to the plane containing d/ and x, and vector m is
lying on the plane.
If the dislocation loop is a smooth curve, C6.25) is written as
C6.26)
after integration by parts, or
., «)d/ C6.27)
with the use of the notation in C6.16). If a vector t is introduced along the line

336 Chap. 6 Dislocations
n = t x m
Fig. 36.4. Dislocation loop L.
xx', thT,jih(m, n) becomes the ji component of the elastic distortion caused by
a fictitious infinite straight dislocation containing points x and x' with
direction t. This distortion is measured at the end points of vector m. The
expression C6.27) is more convenient to use than the Indenbom and Orlov
expression G.17) or the Lothe A967) and Brown A967) expression, since they
contain higher derivatives of the straight dislocation solution.
If the dislocation loop L is a smooth convex planar loop and point x is a
field point inside L, then C6.27) can be reduced to a more convenient form.
Let us denote by a the angle between the tangential direction at a point x' on
the dislocation loop and some datum, and by в the angle between x- x' and
the same datum (Fig. 36.5). Then, we have
d/sin@-a) = \х-х'\йв,
d/cos@-a)= -d|jc —jc'|,
r=|*-x'|sin(fl-a), C6.28)
da = к d/,
dvh = Phda,

36. Stress fields
337
Fig. 36.5. Geometry at dislocation line element d/.
where к is the curvature of L at x' and p is the unit vector directed toward
the center of curvature. Elementary calculus yields
(ft, sin(e-a) + ph cos(d-a)}
de\r) Sm3F-a)
= -Kth csc3(# — a),
where C6.28) is used. Expression C6.27) is written, therefore, as
C6.29)
C6.30)
where
C6.31)
is the ji component of the elastic distortion at the unit distance caused by the
fictitious infinite straight dislocation containing x and x'. The last result
C6.30) has been obtained by Asaro and Barnett A976). When point x is
located outside L, we can have a result similar to C6.30).
For a planar loop, the Indenbom and Orlov expression G.17) reduces to
Brown's formula A967). Simple calculus gives for two dimensions (see Fig.
36.5),
sin(fl-q) 3
Ijc-jc'I 9<
C6.32)

338 Chap. 6 Dislocations
The elastic distortion of an infinite and straight dislocation is inversely
proportional to the distance from the dislocation line to the field point.
Therefore, G.18) is written as
, C6.33)
ji
Applying the operator of C6.32) twice to C6.33), we transform G.17) to
C6J4)
The above expression has been derived by Brown A967) as a refinement of a
similar expression by Lothe A967).
Complicated configurations of dislocations can often be obtained by super-
superposition of angular dislocations. Yoffe A960) has calculated the elastic fields
for angular dislocations in the unbounded material. Yoffe A961), Comninou
and Dundurs A975), and Comninou A977) have obtained similar results for a
half-space.
Elastic solutions for a circular dislocation loop were obtained by Keller
A957), and Kroupa A960) for the unbounded medium and by Salamon and
Dundurs A971) for a two-phase material. Saito A979) has investigated the
effect of a thin-layer on the semi-infinite medium. Elastic behaviors of
disclination loops near a free surface and a two-phase material has been
investigated by T.W. Chou A971), Chou and Lu A972), Chou and Pan A973)
and Chou and Lu A972). Recently, Sekine and Mura A979) have obtained the
elastic field of a dislocation dipole segment. The elastic field of planar periodic
dislocation networks has been obtained by Mura A964), Owen and Mura
A967), Owen A971), and Saada A976). The solution for a helical dislocation
has been obtained by Owen and Mura A967).
37. Dislocation density tensor
The dislocation density tensor of Nye A953) has already been introduced in
G.33). In this section, the tensor will be introduced by a different method.

37. Dislocation density tensor 339
First, consider a dislocation loop L which is the boundary of slip plane S,
Fig. 7.1. The slip b on S introduces a plastic distortion ft?,
pfi dx = -b,nj dS = -b, dSj, C7.1)
where dx = dxl dx2 dx3, dS is the surface element of S, and и is the unit
normal vector of S. Kroupa A962) has called PJj the dislocation loop density
tensor. The Fourier integral representation of fijj is
/?;(*)= Г ?,?(?) exp(z?-*)d?, C7.2)
J — 00
where
Щ(й) = B^) Г ?/(*) exp(-i€- x) dx. C7.3)
•'-00
Since bt is constant on S, C7.3) becomes
Щ(() = -{imY^jnj exp(-ii-*) dS, C7.4)
where C7.1) is used.
Multiplying C7.4) by - i€hlj^, and applying the Stokes theorem, we have
H-x)rk d/, C7.5)
where d/ is the line element of L, and vh is its unit tangent vector. The
equality - i?, exp( - ;'? ¦ лг) = (д/дх/) ехр( - /| • л:) has been used in the deriva-
derivation.
The dislocation density tensor is defined by
ahidx = bivhdl = bidlh. C7.6)
Its Fourier integral expression is
аы(х)=Г ahl(S) exp(i€-x) d|, C7.7)

340 Chap. 6 Dislocations
where
ahiti) = B*r3 П ahi(x)exp(-it-x)dx C7.8)
J — lY^
or
Х-Фн<Н- C7-9)
Comparing C7.5) and C7.9), we have
*Hi=-*HiMP,- C7.10)
When this is substituted into C7.7), we obtain G.33) or
«*/=-«*///./• C7.11)
which is Kroner's expression A955, 1956).
Although the result C7.11) has been obtained for a single dislocation loop,
it also holds for the case of continuously distributed dislocations, where ahi
and fijl are spatial functions. In this case we can interpret
oA/dx=EV*d/, C7.12)
where the summation is taken on all dislocation segments contained in the
infinitesimal cube dx.
The dislocation density tensor ahi expresses the x,-component of the total
Burgers vector of dislocations threading the unit surface perpendicular to the
:cA-direction. Denoting the Burgers circuit by с (see Fig. 10.2), the total
Burgers vector can be expressed as }s«hivh dS which must be equal to the
multiple value of displacement expressed by A0.11). It may also be expressed
by -},.(}/, dxj or -fsehijPfurh dS after the use of Stokes' theorem G.10).
The stress field due to a continuous distribution of dislocations can be
obtained from C7.12) and G.15),
°ij(x) = CijklftlnhCpgmnGkpJx- x')ahm{x') dx'. C7.13)
An isolated dislocation line is a special case when the dislocation density
tensor takes the form of Dirac's delta function. For instance, the dislocation
line shown in Fig. 4.1 is expressed by a33 = b38(x-l)8(x2), P& =

37. Dislocation density tensor
341
i ' ! t IJ I TT1
t HII11ПI hi! I
' 'til
\ , I f} I! 111 [fif f
1 ////////// /i/ /
i ' ! 11!! 11 ]! П {i
I i}!! I! 11! I!! f I
1 .1! 11! 11! F!I\!
I 4 I ! I f I i ! j I I t'l
* * i i I i f f ' ' ' ~ ш
Т77/////////I
111 i 111111 ill
г
.ft*
-•"zz
Fig. 37.1 Distortion is the sum of elastic (lattice) and plastic (non-lattice) distortions.
Ь38(х2)Н(-хг), and the dislocation line in Fig. 4.2 by a3l = b18(x1)S(x2),
For the continuous distribution of dislocations, it holds that
«(i_,=j8y, + j8/, C7.14)
where /?,, is the elastic distortion; the total distortion is the sum of the elastic
and plastic distortions. The plastic distortion does not produce any distortion
among lattice points since it is caused by gliding (slip), while the elastic
distortion is originated in an elastic deformation of the lattice as shown in Fig.
37.1.
Surface dislocation density
When fifj jumps at a surface E, the surface dislocation density tensor is
defined by
C7.15)

342
Chap. 6 Dislocations
Fig 37.2. Uniform plastic distortion around an inclusion Orowan's dislocation loops aei are
created by the misfit.
where /^A) and /ЗДП) are values of fijj in domains I and II on ?, respec-
respectively, and и is the normal vector on ? directed toward domain II from
domain I. The surface dislocation density tensor has been defined by Bullough
and Bilby A956). The formulation of C7.15) is a natural consequence of
C7.11) for discontinuous quantities on an interface.
Figure 37.2 shows a uniform plastic distortion /?? in the matrix II around
an inclusion. Equation C7.15) gives the surface dislocation density tensor,
an = — /
C7.16)
which is defined on the surface of the inclusion. These dislocations are
Orowan's loops A948, 1959), discussed in Section 43.
Another interesting example of the surface dislocations can be found in the
crack problems discussed in Sections 27 and 32. A slit-like crack with length
2аг along the Xj-axis subjected to a202 at infinity was simulated by the
infinitesimally thin elliptical inclusion (a2 -> 0),
x\/a\ + x\/a\ = \. C7.17)
The equivalent eigenstrain ef2 is given by B7.14) and B7.17). This inclusion
can be replaced by the surface dislocations C7.15) where /J?(II) = 0 and
= ?
2
Then, we have
where nx is
1/2
C7.18)
C7.19)

37. Dislocation density tensor 343
from C7.17). From B7.14) and B7.17),
a32 = (l-'K(*i//"»2)«i. C7.20)
and, therefore,
2«32 = 2A - i-)o&(*i/fl,)/ji(l - A/a\f2 C7.21)
after using C7.17) and letting a2 -* 0. The above dislocation distribution
C7.21) defined in \хг\ <аг simulates the stress disturbance of the slit-like
crack under 02°2. The sum of the two surface dislocations defined on the upper
and lower surfaces of the infinitesimally thin inclusion gives the factor of 2 in
C7.21). The distribution of dislocations C7.21) has been suggested by Eshelby
A957); see also Bilby and Eshelby A968).
In the above discussion we have derived the equivalent dislocation distribu-
distribution from the equivalent inclusion for a given crack problem. It is also possible
to find an equivalent inclusion from an equivalent dislocation distribution.
For the Dugdale crack we have
_»» *.(.?-cr+.(.?-,?r ta|jri|<e C7,2)
from C2.37) in the BCS model. The equivalent eigenstrain t%2 is obtained
from C7.18). It is found that a2t22 is zero at xl = ±аъ infinite at xl = +c,
and finite at хг = О.
Impotent distribution of dislocations
Mura A968) has pointed out that if
У=-Р5> C7.23)
the displacement and stress fields caused by ahi as defined by C7.11) or
C7.15) become identically zero. Such a distribution of dislocations is called
impotent. It is obvious from C.23) and C.25) that tfj = tfj = \Щ + ft?) = 0
leads to м, = a,y = 0.
As an example, consider the case
i)- C7.24)

344
Chap. 6 Dislocations
(a)
01,
Pi
(c)
1
(d)
Fig. 37.3. Impotent dislocation distribution.
Equation C7.11) leads to an impotent distribution of dislocations a31(x1) =
-#fi,i(*i)- If P& and Pu are uniform in xx < 0 and zero in x1 > 0, C7.15)
leads to a31 = /?? along the surface jc: = 0 as shown in Fig. 37.3(d). Figure
37.3(a) deforms to Fig. 37.3(b) by jSf1; while Fig. 37.3(a) deforms to Fig.
37.3(c) by PB. The superposition of the deformations due to /}? and /?f2
becomes Fig. 37.3(d). The elastic energy is zero and the distribution of
dislocations in Fig. 37.3(d) results in an impotent distribution of dislocations.
The dislocation distribution shown in Fig. 37.3(b) is the polygonization.
Another example of an impotent distribution of dislocations is shown in
Fig. 37.4, where
C7.25)

38. Dislocation flux tensor
345
э
г—
\
ч.
гИ
Fig. 37.4. Impotent dislocation network.
Planes perpendicular to the Xj-axis rotate clockwise by the dislocation array
a 22, while they rotate counterclockwise by a33. The resulting deformation and
stress fields vanish after the superposition of a22 and a33. This impotent
distribution of dislocations constructs a dislocation network of minimum
strain energy.
38. Dislocation flux tensor
Using the analogy of heat conduction, we consider the growth rate of
dislocations threading an arbitrary surface S; see Fig. 38.1. Let the dislocation
density tensor change by an amount ahi per unit time. The growth rate of the
total Burgers vector of these dislocations is /5аЛ|иА dS, where и is the normal
vector on surface S. The net transport of dislocations moving through the line
element d/ of the boundary S is v(dlx V) от f.lh]Vlvh dlj, where V is the
dislocation velocity and v is the dislocation direction. The growth rate of the
total Burgers vector on S is equal to the flux of dislocations moving through
L, namely,
dh,nhdS=- jelhjVlhldlj,
Li
s
where
C8.1)
C8.2)
The Burgers vector for a single dislocation has been denoted by b in C8.2) and
the summation applies to all dislocations. Mura A963) has defined Vlhi as the
dislocation flux (or velocity) tensor. Applying Stoke's theorem of integration
to the right-hand side of C8.1), it follows that
C8.3)

346
Chap. 6 Dislocations
Fig. 38 1. Dislocations threading surface S increases by moving-in dislocations through L
Since this holds for any surface S, we have
ahi =
C8.4)
On the other hand, the motion of dislocations causes a plastic distortion rate.
For example, /?? is caused by the + .xrdirection motion of a31 and by the
-Xj-direction motion of an, namely
More generally,
When it is substituted into C8.4), we have
or
ahi = ~
C8.5)
C8.6)
C8.7)
C8.8)
by integrating with respect to time. Expression C8.8) is the same as C7.11).
We shall derive C8.6) by a different method, starting from a consideration
of a single dislocation loop whose line element is moving with velocity V. The
plastic distortion associated with this dislocation loop L can be expressed in
the Fourier integral form,
Pfiix, 0= /"Г fifitt, «) exp{i(«-*-«0} d?d<o, C8.9)
J J — oo
where
//°° 0/(x, /)exp{-i($.x-«O}dxd/. C8.10)

38. Dislocation flux tensor 347
Substituting C7.1) into C8.10), we have
i(S,<*)=-Bvy4bir dtf eKp{-i(i-x-at)}dSj(x). C8.11)
•'-oo JS(t)
f
oo JS(t)
When C8.11) is multiplied by -iu, we obtain
-/<оД;(^б)) = BтГ)-46,.Г dtf ^-
•'-oo JS(t)dt
f
oo JS(t)
C8.12)
The first term in C8.12) vanishes by introducing proper initial conditions such
as
5(?) = 0 aW=+oo. C8.13)
The increment of the slip surface S swept by d/ is determined by the vector
product V and d/, that is,
j C8.14)
Then, C8.12) can be written as
/<H* C8.15)
Г /
'-oo JL
or, from C7.6), as
C8.16)
On the other hand, we have from C8.10),
~шЩй, «) = B^) /¦/¦00 0p(x, t) exp{ -i({-x- at)} dx d/. C8.17)

348 Chap. 6 Dislocations
Comparing C8.16) and C8.17), it holds that
#=-W>*. C8-18)
or
#=-«уЛ,. C8.19)
Although all these formulae have been obtained by considering the motion
of a single dislocation loop, the result C8.19) also holds for continuous
distributions of dislocations by the principle of superposition.
The general theory of moving dislocations has been developed by Kosevich
A962-65), Mura A963), Bross A964,1968), Zorski A966), Amari A968),
Gunther A967,1969), Kossecka A969), Lardner A969), Malen A970), and
Minagawa A971). The earliest work on the corresponding field equations has
been done by Hollander A960), where Poisson's ratio has been assumed to be
zero for mathematical simplicity.
Line integral expression of displacement and plastic distortion fields
The elastic distortion and stress fields of a dislocation line L (or loop) are
expressed by the line integrals along L in G.9) and G.15), while the displace-
displacement field is expressed by the surface integral on the slip surface S in G.6).
The displacement field can also be expressed by a line integral, if a creation
history of the dislocation line is prescribed. First, consider thedisplacement
field (8.9) for a given Щх, t), where e* „(?, <o) is replaced by ?fm(|, w),
u,(x, t) = -/Г iCJlmHftM, «Н/^Ш ^D-'U, a)
?-x-uf)} d?dw, C8.20)
where
Д>и(«, Ы) = B»Г7Г #«(*, 0 exp{-/(*• *-«/)} dxdr. C8.21)
•'¦'-oo
If 0? is a plastic distortion rate associated with Vnahi through C8.18), then
C8.20) can be written in terms of Vaahi. From C8.17) and C8.18),
Vk(x, t)ahm{x, t)
o
Xexp{-/(?¦*-uf)} dxdf. C8.22)

38. Dislocation flux tensor 349
When it is substituted into C8.20),
u,(x, t)= -B«)-\khfJjf^CJlmi.Vk(x', t')ahm(x',r)
X | ^'^ ехр{*.(х-х>) - /«(/-/')} d? du> dx' US.
C8.23)
When a discrete (single) dislocation with Burgers vector fe, is considered,
expression C7.6) is used and the integral with respect to x' in C8.23) becomes
a line integral along the dislocation line L. The time derivative of C8.23) is
r dt'CJ[mnVk(x', /')G0,,(x-*', *-/'),
C8.23.1)
where GtJ is Green's function defined by (8.10).
When a uniform motion of the dislocation distribution with a constant
velocity is considered, we can write
ahl(x,t)=ahi(x-Vt). C8.24)
Putting
x'-Vt'=y, dx' = d^, C8.25)
the integration with respect to t' in C8.23) yields 2эт8(? ¦ V-u). Then, C8.23)
becomes
xexp{i?-(x-y- Vt))d?dy. C8.26)
The above result can be directly applied to problems of the static disloca-
dislocation distribution by a limiting process V-* 0 and Vk/V'-* vk, where t> is the

350 Chap. 6 Dislocations
direction of motion w
Then C8.26) becomes
direction of motion when the dislocation ahm is introduced into the material.
C8.27)
It is interesting to note that the displacement field for a given static dislocation
distribution ahm(x) is not unique, depending on the choice of v.
For a single dislocation line L, using C7.6) we have
C8.28)
This expression has been obtained by Mura A971).
The plastic distortion as well as the displacement for a given static distribu-
distribution of dislocations is not a state quantity. It is uniquely determined only when
the history of creation of the dislocations is given. As with the displacement
field, the history is assigned by a given uniform motion of dislocations in the v
direction with an infinitely small velocity V.
Substituting C8.22) into C8.9), we have
For a uniformly moving dislocation distribution with a constant velocity V,
C8.29) becomes
C8.30)
For a static distribution of dislocations, after limiting process V -* 0,
vk/v, C8.30) becomes
idy. C8.31)
Г J-ahi
-oo '*¦ v

35. Dislocation flux tensor 351
It is easy to show that two ftjj with different v yield the same dislocation
density tensor when C8.31) is substituted into C8.8) and the divergency law
«*/.*(*) = 0 C8.32)
is taken into account. Equation C8.32) can be derived from C8.8). It implies
that dislocation lines must close on themselves.
The elastic field of moving dislocations
We start with the solution (8.11), where (.*mn is taken as /?,fm. The plastic
distortion tensor /?y° is related to the dislocation density tensor through C8.8)
and its time derivative is related to the dislocation velocity tensor through
C8.6). Expression (8.11) is
«,(*, /)= - (Г СЛ*.ЛРЛ*', f)G,jj(x-x', t-t')dx'dt'. C8.33)
The distortion is
«,,,(*, t)=- /Г Ск1тЛрЛ*', t')GlkJj(x-x', t~t')dx'dt'
J ¦'-00
= -{Г ^jn^PqhCklmnGlkJp^m + CklmnGlkjJfm) dx' At',
•'•'-00
C8.34)
since tjnif.pqh = 8jpSnq - 8iq8np.
After integrating by parts the first term in C8.34) and applying (9.1) to the
second term, we have
+ PGmi0jrn)dx'dt'. C8.35)
Equations C8.8) and C8.6) are used in C8.35). Then, the elastic distortion
/},,. = ut j - fit is obtained as
|8,,(*, /) = [П {ejnhCklmnGlkJahm + PGm,ejnhVnhm) dx' dt'. C8.36)
•'•'-oo
The corresponding stress is otj = Cjjkl$kl.

352 Chap. 6 Dislocations
For a single dislocation loop L, ahm dx' = bmvh d/ and Vnhm dx' = Vnbmvh
d/ and therefore,
fij,(x, t)= Г dt'f{tJnhCk!mnGikJ(x-x', t-f)
•'-00 JL
+ PGmi{x-x', t-t%nhVn}bmvhdl{x'), C8.37)
where v is the direction of dislocation element dl.
Wave equations of tensor potentials
The following tensor potentials have been defined by Mura A964):
Bkmji{x, t) = jr Gkm(x-x', t-t%1(x', t')dx'dt',
Amnhi(x, t)= [Г Gkm(x-x', t-t')Vnhi(x\ t')dx'dt\ C8.38)
J •'-00
Gkm(x-x', t-t')ahi(x', t')dx'dt'.
Then, the property of Green's function leads to the following wave equations,
Cijkt"kpnm,lj ~ P"ipnm ~ ~°ipPnm'
Cijk^kphm,lj - Ptiphm = ~8ipahm' C8.39)
^¦ijkl-^kpshmjj ~ P-^ipshm = ~"ipVshm-
From the relations C8.8) and C8.19), we have
*kphm hsn kpnm,s>
"~ "kpnm-
C8.40)
The second and third equations in C8.39) are similar to the wave equations of
the potentials derived from the Maxwell equations in the vacuum,
C8.41)

39. Energies and forces 353
where с is the velocity of light, e the charge density, and v is the velocity of
the charge. It is seen from C8.2) that the right-hand terms in C8.41) are
similar to the right-hand terms in C8.39). The Lorentz condition
<j> = 0 C8.42)
corresponds to the relation
ФкрИт= ^kphlmj^ ^kplhm.l C8.43)
which is derived from C8.40) by eliminating B. AkpMml vanishes when Vhlm
can be written as Vhalm with constant Vh, since <x/m/ = 0, where differentia-
differentiations with respect to jc and / of the potentials in C8.38) are reduced to the
differentiations of the source functions (the flux and density tensors) with
respect to x' and t' after integration by parts.
When В and ф in C8.39) are known, the elastic distortion is evaluated from
C8.36),
Рл = ejnh(Ckim&khm.i + pAm,nhm), C8.44)
where p is assumed to be a constant.
The analogy between the dislocation theory and the electro-magnetic theory
has been discussed by several researchers in different contexts, e.g. Hollander
A960), Schaefer A969), Kroner A958), Gunther A967), Minagawa A971)
among others. The gauge invariant associated with dislocations has been
discussed by Golebiewska-Lasota A979), Edelen A980), and Gunther A984).
39. Energies and forces
In this section we consider the elastic strain energy of a static dislocation
distribution, the interaction energy, and the forces acting on dislocations.
The elastic strain energy W* caused by a plastic distortion B.P has the same
form as A3.3),
W*=-\T o.flPdx, C9.1)
•'-00
where /?? is associated with ahi through C8.8).

354 Chap. 6 Dislocations
For a dislocation loop L, C7.1) leads to
jbinjUS, C9.2)
where S is the slip plane bounded by L. For example, the straight screw
dislocation, shown in Fig. 4.1, has an elastic strain energy per unit length of
dislocation line,
W* = J^logMo- = 5?log(Jl/r0), C9.3)
where R is the size of material and r0 is the atomic distance of lattice
(dislocation core radius).
If the straight dislocation is in an anisotropic medium, a,y in C9.2) is
calculated from C6.16). Then, the elastic strain energy per unit length of
dislocation is
Jro klh
t,n)bjnj\og{R/r0), C9.4)
klh
where S in C9.2) has been taken as the plane with a normal vector л as shown
in Fig. 36.1.
The self-force on a dislocation loop is derived from the variation of W* for
an infinitesimal change of /?? in C9.1). The variation becomes
SW*
= - Г оиЩ Ах. C9.5)
J — rr\
The factor \ disappears when a variation of A3.1) is taken and integration by
parts is performed as in A3.2). The variation Sefj is caused by a vertical
displacement 8(- of a line element d/ of the dislocation loop L. It holds from
C8.2) and C8.6) that
8f};, = -ejmMm<*ni- C9.6)

39. Energies and forces 355
Since Sefj = \Щ1 + Щ) and ani dx = vnb, d/, C9.5) becomes
mnvnb,o48iim&l, C9.7)
where v is the direction of the line element.
The Peach-Koehler A950) force acting on the dislocation line element is
defined by
/«=-««я/лА- C9-8)
Then,
f8tmdl. C9.9)
The interaction energy between the dislocation L and a stress field a/} is
defined in the same way as A3.13),
AW=> - Г ofcrdx. C9.10)
•'-oo
Its variation becomes
mdi, C9.П)
where /JJ has the same form as C8.9) with afr The irreversibility of plastic
work done on d/ by atJ and a° is
(/«+/J?K>0, C9.12)
where V is the velocity of the dislocation Jine. The above expression can be
extended to a continuous distribution of dislocations ani, moving with disloca-
dislocation velocity tensor Vmni. The rate of plastic work per unit volume then
becomes
-e [a +a°)V > 0. C9 13)

356 Chap. 6 Dislocations
The last result is important in investigating the mathematical theory of
plasticity in the context of the dislocation theory, described in later sections.
Further investigations on the self-force fm are presented by Asaro and
Hirth A973), Barnett A976), Gavazza and Barnett A976), Minagawa A970),
Das et al. A973), and Golebiewska-Lasota A978), among others. Line integral
expressions of the interaction energy and self-energy of dislocation loops in
anisotropic materials have been obtained by Mura A969).
Dynamic consideration
Let us consider the following Lagrangian functional for a moving dislocation
loop in an infinitely extended material:
= p dtf {{рщщ- HA)d*' C914)
where
<V, = P",> C9.15)
aij ~ CijklPlk-
The above formulae are defined everywhere except on the slip surface S. The
boundary of S (which is the dislocation line) is moving with velocity ?. The
variation of C9.14) becomes
S /Vd/ = /'' d/ f {pufiu, - a,//?,,) Ax
Н*1 C9Л6)
Applying the Gauss theorem, we have
D OXj J
= ( confabs - f ои^щ dx, C9.17)
Jss Jd

39. Energies and forces 357
where SS is the variation of slip surface 5, which is expressed by the vector
product of a virtual displacement of the dislocation position S? and the
dislocation line element v d/,
nj8S = ejlhS^vhdl. C9.18)
In the above derivation, [и,.] is the difference of м, evaluated at the upper and
lower surfaces of the slip plane: [и,] = Ь, on S. After giving a vertical
displacement S?; to the position of the dislocation, the variation [Su,] = S[u,]
is zero on S, and Ъ, on SS.
Furthermore, we have
>ui-r(Sui)dtdx = - f pufoLn SS- (hpu,8ui dt dx. C9.19)
Substituting C9.17), C9.19) into C9.16), we obtain
5 P& d/ = - f' d/$ (рм,.?. + o^Vj/*^/"* d/. C9.20)
/ / Z*
The term puiiJbjejlhvh is called the Lorentz force. Since ?yd? = 8?, and
ej,h8?j8?, = 0, C9.20) becomes
5 \h<e dt = - Г1 бнфоцЬеЛр,, d/. C9.21)
The Lorentz force does not contribute to the energy balance and the result is
the same as the static case C9.7).
The elastic solution of a uniformly moving straight dislocation has been
obtained by Frank A949) and Eshelby A949). Frank has shown that when a
screw dislocation is moving with a constant velocity V, the total energy (the
sum of the elastic strain energy and the kinetic energy) is given by E0/(l —
V2/c2Y/2, where Eo is the elastic strain energy of the dislocation at rest, and
с the shear wave velocity (ju/pI/2. His calculation has been stimulated by the
work done by Kontorova and Frenkel A938) who studied a one-dimensional
dislocation model and found a strikingly analogous form to that of a particle
in special relativity. Frank's calculation was extended to an edge dislocation
by Eshelby A949). In this case no relativistic relation is found since the
dilatational wave velocity is also involved.

358 Chap. 6 Dislocations
The uniform motion of a straight dislocation is possible in the absence of
an applied stress since oijbiijlhvh becomes zero at the dislocation position. It is
not yet clear how the dislocation arrives at the constant velocity. Eshelby has
extended his analysis to the Peierls dislocation A940). No applied stress is
necessary for uniform motion if the width of the dislocation is 2f = {b/(\ -
v)}D(V)/D@) for an edge dislocation or 2f = b[S for a screw dislocation,
where D(V) = -2nBc2/V2)(y - a4/p), у = A - V2/a2)x/1, j8 = A -
V2/c2f/2, a = A - ?2/2с2У/2, с = (/x/p)V2 and a = {(\ + 2jti)/p}1/2. It is
interesting to note that the width of the edge dislocation vanishes when
D(V) = 0, the solution of which is the Rayleigh wave velocity. The width of
the screw dislocation vanishes at V = c. Eshelby's calculation has been ex-
extended by Leibfried and Dietze A949) to a dislocation moving in the middle
of a plate (direction of dislocation parallel to the plate surface). Weertman
A960, 1961, 1966) has completed Eshelby's analysis A949) by calculating the
elastic strain energy and the kinetic energy of the edge dislocation. He has
found that the shear stress on the slip plane (except at the dislocation position)
decreases with increasing dislocation velocity, vanishes at the Rayleigh veloc-
velocity, and changes signs at velocities higher than the Rayleigh velocity. Because
of this, edge dislocations of like sign attract each other (contrary to the usual
situation) when the dislocations are traveling at velocities above the Rayleigh
velocity. Such a critical velocity of a dislocation is called the threshold
velocity. The threshold velocities have been calculated for various crystals by
Teutonico A961, 1962, 1963) and Weertman A962). They started their stress
analysis from the general solution obtained by Bullough and Bilby A954) for a
uniformly moving straight dislocation in an anisotropic media. If a dislocation
of general shape is considered, the formula C8.23.1) is recommended.
The effect of free surfaces and the inhomogeneity of materials on disloca-
dislocation motion has not been fully investigated. Weertman A963) considered a
dislocation moving on the interface between two isotropic media of different
elastic constants and densities. The case where a dislocation is moving parallel
to the interface has been solved by M.S. Lee A972).
Although the energy of a moving dislocation becomes infinite as its velocity
approaches the sound velocity, there are dislocation solutions of the elastic
equations for greater dislocation velocities than c. This dislocation-like solu-
solution was first demonstrated by Eshelby A956). He suggests its possible
application to the propagation of diffusionless transformations by disloca-
dislocations; however, no successful application has been reported. Callias and
Markenscoff A980) analyzed the stress field due to the general transient
motion of a supersonic screw dislocation and in particular the wave-front
behaviour.
The applied stress required for the supersonic dislocation motion becomes

39. Energies and forces 359
infinity. This difficulty is avoided by Weertman A967, 1969) by considering a
spread dislocation with a distribution function B(x1 — Vt). If В is a constant
in the region —A<(xl — Vt) <A and zero elsewhere, the equilibrium condi-
condition is satisfied for a uniformly applied stress. Weertman also has discussed
atomic force law under which the equilibrium condition is satisfied without
applied stress: see the review paper by Mura A972). For a spread supersonic
dislocation, Clifton and Markenscoff A981) estimated that the required stress
to sustain the supersonic velocity would exceed the theoretical strength of the
material.
Almost all calculations for moving dislocations have been limited to straight
infinite dislocations. Very few calculations have been done for circular disloca-
dislocation loops. Giinther A968) has obtained the elastic solution for a circular edge
dislocation moving uniformly in the direction of the normal to the circular
plane. In the region V<c <a the elastic field shows double Lorentz contrac-
contractions and in the region с < a < V there arise two Mach cones radiated from
the dislocation loop with angles +sin~1(F/c) and +sin~1(F/a) with respect
to the direction of the motion. No disturbance is seen outside the domain
bounded by the cones and the circular plane, where с and a are the shear and
dilatational velocities.
Unlike the uniform motion of a straight dislocation with a subsonic
velocity, an oscillating straight dislocation generally requires the application of
an oscillating stress (forces vibration). Free vibrations, however, are possible
under special circumstances. By using Eshelby's elastic solution A949) for an
oscillating screw dislocation and the equilibrium condition, Nabarro A951)
has obtained the amplitude and phase-difference for the position of the
oscillating dislocation as functions of the amplitude and wave number of an
incident sound wave. He also has calculated the scattering cross-section (the
ratio of the rate of radiation to the incident energy flux). Same calculation is
done for an edge dislocation by Kiusalaas and Mura A964). On the other
hand, Leibfried A950) has pointed out that a dislocation moving through a
flux of sound waves (which are a natural consequence of the thermal energy of
the crystal) experiences a retarding force which is proportional to the disloca-
dislocation velocity. The numerical value of his result, however, is criticized by
Nabarro A951), Lothe A960), and Eshelby A962). Eshelby has used a kink
model for mathematical simplicity and found that a kink moving through an
isotropic flux of elastic waves has a scattenng cross-section proportional to the
square of its width, and it experiences a retarding force proportional to the
product of its velocity and the energy density of the waves. Pegel A966), Laub
and Eshelby A966), and Ninomiya and Ishioka A967) have obtained disper-
dispersion curves for the free vibration of a straight dislocation. They found that free
vibrations are possible for a certain range of frequencies and wave lengths, but
that in general a suitable applied stress is required to maintain the vibrations.

360 Chap. 6 Dislocations
Eshelby A953) has considered several cases of the accelerated motion of a
screw dislocation. The dislocation is at rest for negative t, and thereafter
moves 1) with a velocity c2t/(x\ + c2t2)l/2 and acceleration c2x\/(x\ +
c2t2K/2, 2) with a constant acceleration, 3) with a constant velocity. He has
calculated the uniform applied stresses required for these motions. He also has
calculated the velocity of the dislocation when a constant stress is applied at
t = 0 and t = t1 so that the total impulse is a constant value. Like Nabarro
A951), Eshelby also accounts for the Peierls stress law. According to the
present author's opinion, no such attention is necessary as long as the
equilibrium condition is being considered at xx = 0, x2 = \b. Some details of
the elastic field caused by the uniform motion of a dislocation starting from
rest are given by Ang and Williams A959) and by Kiusalaas and Mura A964).
The velocity field has a singularity at the wave front. However, if the
dislocation starts to move with a constant acceleration, the velocity at the
wave front becomes zero. A solution for the general non-uniform motion has
been obtained by Markenscoff A980) for a screw dislocation and by
Markenscoff and Clifton A981) for an edge dislocation. The self-stress of a
dislocation starting from rest and moving with a constant velocity has been
obtained by Kiusalaas and Mura A964) for a screw dislocation and Clifton
and Markenscoff A981) for an edge dislocation. Recently, Markenscoff and Li
A984) obtained the exact solution for the field radiated from a circular
dislocation loop expanding in its own plane with constant velocity.
The interaction between a varying stress field and a moving screw disloca-
dislocation has an electro-magnetic analogy as shown by Eshelby A953) and Nabarro
A951). From this analogy, Nabarro predicts that a moving dislocation receives
a Lorentz force (in addition to the Peach-Koehler force) similar to a moving-
line density of charge. Since the Lorentz force has been predicted strictly from
this analogy, its existence has been criticized by several authors. Although in
some papers the Lorentz force is derived from a Lagrangian formalism, either
the mathematics or the physics involved in these derivations is questionable.
The scattering of elastic waves by a dislocation is only one of the many
energy dissipative mechanisms which have been suggested by a number of
investigators. Eshelby A949) has shown that the thermoelastic effect around a
moving edge dislocation produces an irreversible heat flow causing energy
dissipation. A more rigorous analysis of the thermoelastic dissipation due to
an edge dislocation moving at an arbitrary speed has been performed by
Weiner A958). In order to produce a similar thermal effect for a moving screw
dislocation, Mason A960) has proposed a shear wave effect which would
instantaneously raise the temperature of those phonons which have compo-
components along the compressed direction, while lowering the temperature of those
which have components along the extended direction.

40. Plasticity 361
As an element of the dislocation line moves through the lattice, its potential
energy varies in a roughly sinusoidal manner; Orowan A940). Therefore, the
traveling dislocation experiences an oscillatory force. The energy is dissipated
through this oscillation by a mechanism similar to that mentioned in the last
section. Hart A955) has calculated the magnitude of this dissipation. Various
other dissipative mechanisms have been studied by Lothe A962), Koehler
A952) and Granato and Liicke A956). Some atomistic aspects of thermally
activated motions of dislocations have been investigated by Weiner A969).
Kroner A955), Hollander A960), Schaefer A969), Giinther A967), Minagawa
A971), Golebiewska-Lasota A979) among others discussed some analogies
between the electromagnetic and moving dislocation fields. However, no
practical applications have been found yet.
40. Plasticity
In this section the mathematical theory of plasticity is interpreted in terms of
dislocation theory. The basic idea is that a stress field in continuum plasticity
is the sum of the applied elastic stress (solution of a boundary value problem
in elasticity) and the dislocation stress caused by dislocations created in the
so-called plastic domains. The Mises yield criterion and the Prandtle-Reuss
relation between stress and plastic strain rate are also derived from the theory
of continuous distributions of dislocations.
Mathematical theory of plasticity
The fundamental equations of continuum plasticity are reviewed here before
the dislocation theory is introduced. In the absence of body forces the
equation of equilibrium is
oUJ = 0. D0.1)
The traction boundary conditions are
aijnj = F, on S, D0.2)
where Ft is the applied force on surface S and ni is the exterior unit normal
on S. The strain etJ is the sum of the elastic strain ei} and the plastic strain ef,
c,y = e,.. + ?fr D0.3)

362
Chap. 6 Dislocations
(о)
(b)
Fig. 40 1 Stress-strain curve in the simple tension test (a) is perfectly elasto-plastic material, (b)
is work-hardening material
The strain e,- is compatible and expressed by
D0.4)
where м, is the displacement component. The material is in a plastic state
when
!„ „ _ u2 _ 1=2
2 и О ~ ~ 3 '
where
Jkk
D0.5)
D0.6)
is the reduced stress component (or deviatoric). к is the shearing yield stress
for a perfectly elasto-plastic material and the shearing flow stress for a
work-hardening material, while a is the yield stress or flow stress in a simple
tension test, a (or k) is a constant for the perfectly elasto-plastic material (see
Fig. 40.1a), while it is a function of plastic strain for the work-hardening
material (see Fig. 40.1b). In this latter case we have
о = а
where
D0.7)
D0.8)
is the effective plastic strain increment, a is also called the effective stress and,
from D0.5),
a =
X
D0.9)

40. Plasticity 363
The condition D0.5) is called the Mises yield criterion A928). Finally, the
constitutive equations are
dekk = dakk/3K, D0.10)
where ц is the elastic shear modulus, and К the bulk modulus, dju* is an
unknown sealer quantity for the perfectly elasto-plastic material. For the
workhardening material, it is
d;U* = 3 dep/2a = 3(dey do) da/2a. D0.11)
The last result has been obtained from defj dtfj = (djn*J.y,7.y,v and definitions
D0.8) and D0.9). The slope do/ dlp is obtained from the stress-strain curve in
Fig. 40.1(b). The third equation in D0.10) has been proposed by Reuss A930).
It is a modification of the Levy A871) and Mises A928) equation, de,- = dju*.s,
A special case of the Reuss equation has been obtained by Prandtle A924) for
the plane problems.
Dislocation theory
Using the dislocation flux tensor Vnhi in C8.19), we write the plastic strain rate
as
where
#=-«,Л,. D0.13)
The rate of plastic work per unit volume is
SW/dt = OiJifj = -е/л,а„КМ1.. D0.14)
The plastic deformation does not cause any volume change, that is,
?"?=-«/Л.- = 0, D0.15)

364 Chap. 6 Dislocations
since the plastic work is irreversible,
Mura A965) has shown that the two conditions D0.15) and D0.16) are
satisfied when
Vlhl = dti*glhi D0.17)
with
&,ы=-\ЧнЯ» D0-18)
where d^* is a positive sealer quantity. glhi is the x, -component of the force
acting on ahi dislocations. The magnitude of the force must be a constant к
for the perfectly elasto-plastic materials. Equation D0.13) leads to
D0Л9>
which is the Mises yield criterion D0.5)
When D0.12) and D0.13) are used with D0.17), we have
D0.20)
which corresponds to the Reuss equation in D0.10). d/z* may be determined
from the simple tension test for work-hardening materials as seen in the
mathematical theory of plasticity, dju* is a history-depending quantity. Kroner
A963) has assumed it to be a function of dislocation loop density. The
constitutive equations in plasticity have been further studied by Eisenberg
A970), Hahn and Jaunzemis A973), Lambermont A974), Smith A970), Kari-
haloo A975), Bodner A978), Weng and Phillips A978), and Werne and Kelly
A978), among others, from the viewpoint of continuous distributions of
dislocations. Gilman A966) has discussed the stress-strain law in terms of
multiplication and motion of dislocations.
After a series of original works by Nye A953), Bilby, Gardner and Smith
A958) and Kroner A963), Mura A967) has postulated that the stress field in
continuum plasticity is the sum of the applied elastic stress and the dislocation
stress. He also has shown that the dislocation density tensor ahi can be
expressed in terms of the stress components if the impotent dislocations are
excluded, and the material is infinitely extended.

40. Plasticity 365
We start our discussion with C7.14), that is,
«,-,, = ?,+ 0/. D0.21)
When substituted into C7.11), we have
«A, = € A/A'.' = ?*'Л<-.' + «А/у"//./' D0.22)
where
?, = <?,, + «,,.. D0.23)
The first term in D0.23) is the elastic strain and the second term is the
rotational tensor («,,- = —«,-_,)- The part of <xhi contributed from the second
term in D0.22) is impotent. The displacement and stress fields due to «/^w,,,
can be considered to be equivalent to those caused by eigenstrain /?* = - «,,
[see C7.23)]. Therefore, the potent part of ahi can be written as
«*/ = W,/./- D0-24)
Since ejt = etj = Q^namn, D0.24) becomes
«a, = C,;l€A/jamn,,. D0.25)
The above result indicates that the potent dislocation distribution can be
obtained if the corresponding stress distribution is known.
The basic idea of the dislocation theory in connection with the continuum
theory of plasticity is that a stress field in an elasto-plastic medium is the sum
of the (potent) dislocation stress and the elastic applied stress. Several exam-
examples are shown in the following subsections.
Plane strain problems
Consider an isotropic material subjected to an applied boundary force. The
material is deformed to an elasto-plastic state of plane strain. In the plastic
domains, the sum of the applied elastic stress and the dislocation stress
satisfies the Mises yield criterion,
Цох-оуJ + ох2у = к2. D0.26)

366 Chap. 6 Dislocations
When the material is perfectly elasto-plastic, the stress field satisfying the
above condition may be written as (Hill 1960)
ax= —p — к sin 2<J>,
ay= -p + k sin 2ф, D0.27)
axy = к cos 2ф,
where p = — j(ax + ay) and ф is the angle between the direction of the first
shear line and the x-axis measured counterclockwise. On the first and the
second shear lines, the shear stress takes on the value of + k.
The stress is related to the elastic strain by Hooke's law in any domain
(plastic or elastic),
ex = sx/2ii-p(l-2v)/E,
D0.28)
where
D0.29)
Sxy °xy J
p=-\{ox + oy).
Equation D0.24) leads to
a,, = berv/bx — der/dy,
У D0.30)
The dislocation density is obtained from D0.30), D0.28), and D0.27). The
components of ahi in the directions of the shear lines 1 and 2 become
_ к дф 1 — 2v dp
1 ju 3s2 E ds2'

40. Plasticity 367
where ds^ and ds2 are line elements of the first and second shear lines. The
above result has been obtained from D0.25) by assuming the x-direction to be
the first shear line and the j-direction to be the second shear line. The Burgers
vector of аг has the direction of the first shear line and that of a2 is in the
second shear line.
The equations of equilibrium with D0.27) lead to the Hencky relations,
as2 os2
Then, D0.31) becomes
0B ~
E
2B - v)
The above result has been derived by Mura A967). A similar result has been
obtained by Eisenberg A970) and Weng and Phillips A976).
Lines of discontinuity in the continuum theory of plasticity are defined as
those lines across which the stress varies in a discontinuous manner. In the
dislocation theory, the lines of discontinuity are the lines along which surface
dislocations are distributed. Excluding the impotent dislocations, we write the
surface dislocation density tensor as,
[eki]n,, D0.34)
«3, = « 31k
where [eki] = eki (in domain 2) - eki (in domain 1), and the normal n, is
directed toward domain 2 from domain 1. The above equation has been
obtained from C7.15), D0.21), and D0.23). The traction force should be
continuous on the line of discontinuity,
k>, = 0. D0.35)
Equations D0.34) and D0.35) lead to
2-"r i
«3i = —р—[Р\п2>
2_v< D0.36)

368
Chap. 6 Dislocations
Fig. 40 2. Dislocation distributions in the fan-shaped plastic domains.
The resultant of ahi along the line of discontinuity is
dx1 dx2 2-v,
ds
X32
ds
-[Ph
D0.37)
where ds is the line element along the line of discontinuity, and dx^/ds - -n2,
dx2/ds = nv The Burgers vector of <xs is tangential to the line of discontinuity
(see Fig. 40.7).
Consider a half-space with a uniform pressure P = B + tr)k along \x\ <a,
v = 0. The first and second shear lines are shown by the solid and dashed lines
in Fig. 40.2. The distribution of dislocations is easily found from D0.28). In
the fan-shaped region in the right-hand side in Fig. 40.2, Эф/Э.^ = l/r, and
as =
_ B-v) 2k
D0.38)
Similarly,
a, =
B-v) 2k
—, a2 = 0,
D0.39)
in the left-hand fan-shaped region. In the above equations, r is the distance
measured from the edge of the indenter.
In order to demonstrate that the stress field is the sum of the applied elastic
stress and the dislocation stress, we consider the simpler example shown in
Fig. 40.3. The half-infinite domain is loaded by a semi-infinite band of

40. Plasticity
369
Fig 40.3 A semi-infinite band of uniform pressure P on the surface of a half-space
uniform pressure P on the surface. The applied elastic stress components are
orA=
•п
D0.40)
<= 2^A + cos 20),
(see Timoshenko and Goodier 1951), where the polar coordinates (г, в) are
used. Let us assume that the plastic domain is — 60 < в < в0. Analogously with
D0.39), the dislocations are distributed along the radii with the density
inversely proportional to r,
a3r=b/r.
D0.41)
The stress field of the distributed dislocations can be obtained by integrating
over the plastic domain | в \ < 00 the solution for a single dislocation (Dun-

370 Chap. 6 Dislocations
durs and Mura A964)) multiplied by the density D0.39). The result is
D0.42)
oe=^~2f,
where
f{6) = B0O + sin 20o)(sin 26 + 2в)-тт sin 2в0 cos 20 + 2-n60
for -\m<6< -0O,
fF) = B6O + sin 2<?0)(sin 26 + 2в) + it cos 20O sin 26 - 2тт6
for -60<в<60,
f{9) = B0O + sin 20o)(sin 20 + 26) + it sin 20O cos 26 - 2тт60
for 60< 6 <\m, D0.43)
and /' = d// d0.
The total stress field is
or = arA + orD,
°e = oeA + of, D0.44)
°re = °ri + <>%¦
The Mises yield criterion becomes
\{or-oef + o^<k2. D0.45)
The equality should hold in —в0 < в < в0, and the inequality should hold in
the rest of the body. The condition D0.45) is satisfied if (Mura 1968)
0),
D0.46)
P = 2k{26a + sin 26O + tr cos 20O)/A + cos 26O).

40. Plasticity
371
Fig 40 4 Dislocation distributions at tht ..Kick tip.
The above relations determine the plastic domain and the necessary distribu-
distribution of dislocations for a given P. The relation between p and в0 and the
stress field D0.44) agree with the classical solution obtained by Sokolovsky
A950).
Vilmann and Mura A979) have considered a two-dimensional distribution
of dislocations as shown in Fig. 40.4, and have modified the Griffith fracture
theory for application to ductile materials. From slip line theory, the stress
field in the fan-shaped plastic domain is
au = к{A + Зчг/2 - 26) - sin 20},
D0.47)
a12 = к cos 26.
From D0.24) and D0.47) the dislocation density in the radial direction is
\-v2k
a, =
D0.48)
where r is the distance from the vertex. The Burgers vector of ar is the radial
direction. The application of this result to fatigue has been considered by
Mura and Vilmann A981).
When the applied load is reduced to zero, an additional dislocation distri-
distribution, Fig. 40.5(b), is superimposed on the old distribution, Fig. 40.5(a).
When the load is applied again, the resulting dislocation distribution is the
sum of those shown in Figs. 40.5 (a), (b), and (c). The sum of the dislocation
distributions (b) and (c) is a dipole distribution which can be observed in
fatigue tests. The second unloading adds -the distribution (b) to the last one
and the subsequent loading adds the distribution (c) to the preceding one and
so on. Thus, the strength of the dipoles increases by the repeating loading.
After some number of cyclic loadings, the crack tip advances by Sa as shown
in Fig. 40.6. The domain of the dipole distribution increases with the growth
of the crack.

372 Chap, 6 Dislocations
(o) a3rMi-i/JK>r
AY
A H 1 У
ХЛ 'У
УУЧЛХ
> v H A x
A H у
(c) a3rMI-i'Lk//i.r x)y
Fig. 40.5 Dislocations distributions at the crack tip. (a) At the maximum stress in the first
loading (b) Additional dislocations created by the first unloading The resulting distribution is
the sum of (a) and (b). The dislocation distnbution at the second loading is the sum of (a), (b)
and (c) The sum of (b) and (c) is a dipole distnbution.
A ^ V У
A H V x
"ты* So %^x So \Hy
.^t*«.
y -у -i л
Fig. 40.6. Dislocation distributions of a propagating fatique crack.

40. Plasticity 373
Fig. 40.7. Surface dislocations are on the lines of discontinuity.
Figure 40.7 shows surface dislocations along the lines of discontinuity when
the wedge becomes fully plastic under pressure P. As seen in Prager and
Hodge A951), we have
[p] = -2ksiny D0.49)
along AOC and DOB, where 2y is the wedge angle. From D0.37) the density
of the surface dislocations is obtained as
a,= ^?-2ksmy. D0.50)
These distributions of dislocations shown in the above examples have been
obtained from the statically admissible stress fields which are essentially
independent of the displacement and strain fields (kinematically admissible
fields). However, the present results of dislocation distributions give possible
patterns of plastic deformation. Knowing the source of the dislocations (crack
tips, for instance), one can evaluate possible plastic strains. These strains are
the product of the dislocation density and the distance from the source.
Beams and cylinders
Nye A953) has introduced the concept of a dislocation density and has
expressed the curvature of a beam caused by dislocations in terms of the
dislocation density tensor. He has considered the case where the dislocation

374 Chap. 6 Dislocations
stress disappears. Read A957), and Bilby, Gardner and Smith A958) have
dealt with the case where the stress does not vanish. They have obtained a
relation among the stress gradient, curvature, and the dislocation density
tensor. In this book, a somewhat different approach is used as illustrated by
the following examples.
Consider an infinitely long beam in the л:,-direction with a height of 2 h.
Under pure bending, all quantities depend only on the .x2-variable, and only
the normal stress and strain in the .Xj-direction are considered. From D0.13)
this shows that all dislocations are of the edge type with the Burgers vector in
the Xj-direction and that their motion takes place in the x2-direction, i.e., the
plastic domain expands in the x2-direction. Although this motion of disloca-
dislocations is of a climb type, it can be regarded as the sum of two slips of two
dislocations whose Burgers vectors are +45° with respect to the xx -direction.
The sum of the two Burgers vectors is equal to the Burgers vector in the
xrdirection.
When the edge dislocations with density bl are distributed uniformly in a
strip domain с < x2 < h in the infinitely extended material, the dislocation
stress is, from D.13) and D.14),
^- Г
-p)J-
2,A-,)./_^4 {-2+p2)
-?\2
dx'-,
= -Y^-(h-c) forx2>h,
1 ~ " D0.52)
= -~-{2x2-h-c) for h>x2>c,
= jZLp(h-c) iorc>x2,
where x = xx — x[ and у = x2 — x'2. The other stress components disappear. If
the dislocations are distributed in the two strips as shown in Fig. 40.8(a), the
stress field becomes
оп = —.т~2—~(*2~с) iorh>x2>c,
аи=0 for с > | jc2 |, D0.53)
— 2иЬ, .
аи = — (х2 + с) ют-с>х2>-п.

40. Plasticity
375
1U111
,111111
111111,
11 1111
111111 111111
111111 X 1 1 1 1 1
Fig. 40.8. Beam under plastic state.
Although D0.53) has been constructed from the solution for the infinite body,
it can be used as the solution of the beam with the plastic domains h > \x2\ >c
if the linear stress
au = Mox2/I
is added to D0.53), where
Mo =
- cfBh
- v),
D0.54)
D0.55)
The stress D0.54) is determined such that the sum of D0.53) and D0.54) yields
zero moment. This must be true since the beam is free from external tractions.
When the beam is perfectly elasto-plastic with the yield stress a0 and is bent
by moment M (applied stress Mx2/I), the total stress must be ±a0 in the
plastic domains, that is,
c)
Mx2
= ±a0.
The above conditions are satisfied when
Mo/I + M/I =
-v) = a0,
-v),
= a31 =
31
v2)aQ/cE.
D0.56)
D0.57)

376 Chap. 6 Dislocations
Then, the total stress becomes the well-known distribution as shown in Fig.
40.8(b), and the moment necessary for this plastic state is
M= 3-aoC/i2-c2) D0.58)
which agrees with the classical solution.
The dislocation distribution shown in this example can also be derived from
D0.25). From the stress distribution in Fig. 40.8(b), a31 becomes zero in the
plastic domains, and equals -ao/cE in the elastic domain. If the uniform
distribution ao/cE is superimposed everywhere, the resulting distribution
becomes identical to the one shown in Fig. 40.8(b). The uniform distribution
does not cause any stress field. E is Young's modulus. The factor A - v2) in
the a31-expression in D0.57) arises from the deviation of the beam theory from
the plane strain state.
Similar calculations can be done for the torsion of a solid circular cylinder
(Mura 1970). Assuming that all dislocations are of the same sign and parallel
to the x3-axis (the axis of the cylinder), D0.13) leads to
233' D0.59)
or, equivalently, to
0gp3=Vr33. D0.60)
This shows that the plastic deformation is caused by the radial motion of a33.
In Fig. 40.9 screw dislocations a33 = by are distributed uniformly in the plastic
domain. The dislocation stress is obtained from D.8). Integrating for this
dislocation distribution we have
oez=-iib3{r2-c2)/2r ioxc<r<a,
D0.61)
= 0 for r < с
If the cylinder is free from external tractions, the following stress must be
added to D0.61) so that the total twisting moment disappears:
oiz = Mor/J, D0.62)
where
D0.63)

40. Plasticity
311
Fig. 40.9 A solid cylinder with radius a under torsion Screw dislocations are distributed in
plastic domain с < r < a.
When the cylinder is a perfectly elasto-plastic matenal with the yield shear
stress k, twisted by moment M, the total stress must be equal to к in
с <r<,a, that is,
c<r<a,
D0.64)
where D0.61) has been approximated by -цЬ3(г — с) by assuming с ~ а » r.
The above condition is satisfied when
Mo/J + M/J =
fib3c = k.
D0.65)
The second equation in D0.65) determines the dislocation density b3, and the
first equation yields M necessary for having plastic domain с <r <a. The
total stress distribution under M is shown in Fig. 40.9. According to the
elementary strength of materials, the torsion of a circular cylinder is described
by
= Or
D0.66)
or
D0.66.1)

378 Chap. 6 Dislocations
Equation C7.11) leads to the dislocation distribution
аи=а22 = в. D0.67)
It can be proved that the dislocation distribution D0.67) is equivalent to
a33 = — 20 if the impotent distributions of dislocations are superimposed on
D0.67). The impotent distributions are
and
ft'2="tol' D0.69)
Then, D0.68) gives
«и = -9,
«зз= -
D0.70)
зз
and D0.69) leads to
a-,-, = — в,
Z
Therefore, the sum of D0.67), D0.70), and D0.71) yields
«зз= -20 D0.72)
which is the dislocation distribution assumed after D0.60).
When the screw dislocations distributed along a circle oscillate radially, all
dynamic responses vanish. This was found by Mura A972) and confirmed
later by Parnes and Beltzer A984).
Shioya and Shioiri A976) have explained the non-uniform yield-zone pat-
pattern in twisted mild steel bars with circular cross-sections in terms of continu-
continuous distributions of dislocations. Flinn A970) and Jimma and Kawai A979)
have investigated the deformation mechanism in fabrication processes by
continuous distributions of dislocations. Jimma and Masuda A977) have
studied longitudinal and torsional waves in a linearly work-hardening material
by using moving dislocations. Smith A958) has proposed a model of a planar

41. Dislocation model for fatigue crack initiation 379
network of moving dislocations at shock fronts in explosive metals. Ohnami et
al. A972) and Shiozawa and Ohnami A974) have studied flow stresses of
metals and X-ray diffraction by means of the geometrical property of continu-
continuous distributions of dislocations. An electron microscopic study of dislocation
walls has been done by Yamamoto, Yagi and Honjo A977). Owen A971) has
assumed the stress of a dislocation dipole to be the work-hardening in the
material and has derived the corresponding stress-strain relations of composite
materials.
41. Dislocation model for fatigue crack initiation
The initiation of fatigue cracks is one of the most important stages in the
fatigue fracture process of metals. A large number of metallographic observa-
observations have been carried out to elucidate the micromechanisms responsible for
crack initiation. The state of the art is found in the recent excellent review
articles by Grosskrentz A971) and Laird and Duquette A972). The site of
crack initiation vanes depending on the microstructure of the material and the
applied stresses. Among the possible sites of crack initiation, the persistent slip
band is a preferential one for pure, single-phased metals and some poly-phased
metals under a low-strain cycling. The cyclic strain is concentrated along the
slip band, accompanied by extrusion or intrusion. Laird A976), Brown A977),
Mughrabi A980), Neumann A983) among others have investigated the nature
of microscopic mechanisms of persistent slip bands.
Most of the models proposed to account for the formation of extrusions
and intrusions, are based upon the Mott assumption A958) that dislocations
move along different paths under forward and reverse loadings. These models,
however, cannot explain the most important experimental fact in fatique: the
'ratcheting' of plastic deformation by a cyclic loading. Lin and Ito A969) have
proposed a model for ratcheting in which a specimen is subjected to a special
initial stress state. Zarka, Engel and Inglebert A980) also have proposed a
phenomenological model for the ratcheting. Recently, Tanaka and Mura
A981) have proposed a more reasonable model for ratcheting by considering
two adjacent layers of dislocation pile-ups. Each layer has a different sign.
Their dislocation dipole model can also yield the Coffin- Manson type law for
the crack initiation and the Petch type equation for the grain size dependency
of fatigue strength. This section will introduce the theory of Tanaka and Mura.
Let us consider an applied shear stress pattern as shown in Fig. 41.1.
Figures 41.2 and 41.3 show models of dislocation pile-ups. Dislocations are
piled up on layer I by the loading (forward) force and on layer II by the
unloading (reverse) force. The signs of the dislocations are opposite in the two

380
Chap. 6 Dislocations
о О
2n-l
V
г
V
4
V Tirr
Fig. 41.1. Applied shear stress pattern.
Surface
Vacancy Dipole
(a) Extrusion
(t>) Intrusion
(c) Extrusion- . , ,
Intrusion " u
Pair
Grain Boundary
Fig. 41.2. Dislocation distribution in a most favorably oriented grain and the formation of
extrusion and intrusion.
Grain Boundary
Fig. 41.3. Dislocation distribution in a most favorably oriented grain.

41. Dislocation model for fatigue crack initiation 381
layers. The dislocations in Fig. 41.2 are created at the surface of the specimen
and moved to the interior of a grain. The dislocations in Fig. 41.3 are created
inside a grain and moved to the grain boundary.
We calculate the plastic strain, number of dislocations, the associated
energy, and the stress field caused by the dislocation motion. Under the first
loading Tj greater than the frictional stress k, the dislocation distribution with
density D^{x) is produced on layer I as shown in Fig. 41.3. (The following
analysis is developed for the dislocation pile-up in Fig. 41.3, but the result also
applies to the case shown by Fig. 41.2 when the plane x = 0 is assumed to be a
free surface.) By assuming к to be constant, we express the equilibrium
condition of dislocations as
TlD + rl-k = 0, D1.1)
where rf is the dislocation stress (the back stress) given by
The domain of the dislocation distribution is -a <x <a, and
(l-p), D1.3)
where b is the Burgers vector, ц the shear modulus, and v Poisson's ratio. The
solution of D1.1) is found by the method in Appendix 4. When the solution is
assumed to be unbounded at x = + a, it becomes
*>i(x) = (т, - к)х/тА(аг - x2f/2. D1.4)
The total number of dislocations existing between x = 0 and a is
D1.5)
[
•'o
The average plastic strain caused by these dislocations is
у, = Г bDx{x)x dx = (Tl - k)ba2/2A. D1.6)
* -a
The multiplication of the number of pile-up layers in a unit area by the Yi
value yields the macroscopic plastic strain. For simplicity, Yi is called the
plastic strain.

382 Chap. 6 Dislocations
The stored energy of dislocations is
r1DbD1(x)x Ax = iy,(Ti - k). D1.7)
When the applied stress is reversed from тг to т2, the dislocations with
opposite signs are piled up on layer II as shown in Figs. 41.2 and 41.3. We
assume that the dislocations in layer I do not move by the unloading. This
assumption of the irreversibility of dislocation motion will be discussed later.
By denoting the density of dislocations on layer II by D2(x) and the back
stress due to D2(x) by т2о, we express the equilibrium condition in layer II as
т2д + т/> + т2 + A; = 0. D1.8)
The distance between layers I and II is assumed to be very small compared
with the pile-up length. Then, rf on layer II can be regarded as the same as on
layer I. Equations D1.1) and D1.8) yield
т2°-(Ат-2к) = 0, D1.9)
where Ат = т1 — r2. Only when 4т is larger than 2к can the dislocations on
layer II start to move from x = 0 and pile up at x= ±a. The dislocation
density D2(x), the total number of dislocations between x = 0 and x = a on
layer II, and the plastic strain increment y2 are obtained as
D2(x)= -(Ат-2к)х/ттА(аг-х2I/2,
N2= -{Ai-2k)a/mA, D1.10)
V2= -{Ar-2k)ba2/2A.
The stored energy of dislocations D2(x) becomes
U2=-\y2(Ar-2k). D1.11)
The pile-up of opposite dislocations on layer II causes a positive back stress on
layer I. This back stress enhances the pile-up of dislocations on layer I during
the next loading. Similarly the dislocations on layer I enhance the pile-up of
dislocations on layer II during the subsequent unloading. This is the ratcheting
mechanism proposed by Tanaka and Mura A981). After repeated loading and
unloading, the extrusion or intrusion is created on the surface of the specimen
as shown in Fig. 41.2.
The increment of dislocation Dk(x), the dislocation number Nk, the plastic
strain increment yk, the back stress increment тк, and the stored energy Uk at

41. Dislocation model for fatigue crack initiation 383
the k-lh step of the loading and unloading processes are obtained in a similar
manner. They are
)() k () yk = (-l)k+lAy,
D1.12)
TkD = {-\)k+1Bk-Ar), Uk = AU,
where
At = Tj — т2,
AD(x) = (At - 2k)x/-nA{a2 - x2f/2,
АЫ=(Ат-2к)а/тгА, D1.13)
Ау = (Ат-2к)Ьа2/2А,
The index к takes 2 и at the minimum stress after n cycles and 2 и + 1 at the
maximum stress after n cycles. The total values of these quantities obtained in
D1.12) or D1.13) are
Ni= E N2n+\ = N\ + nAN, D1.14)
n
for the dislocation pile-up on layer I, and
n
11 V / '— Lmd 1 n\/ — г1?Л UI л I,
n
D1.15)
/1 = 1

384 Chap. 6 Dislocations
for the dislocation pile-up on layer II. D}(x) is written as
?>!(*) = Г,;сА<4(я2-х2I/2, D1.16)
where
D1.17)
The stress field due to the dislocation distribution D^x) is the same as that
for the shear crack under shear stress Tx (see Bilby and Eshelby, 1968). The
stress intensity factor at the pile-up tip is
К, = ГДтгаI72 = {Tl - к + n(Ar - 2k)}{va)W2. D1.18)
Similarly, the stress intensity factor at the tip of layer II becomes
Kn = Tn{va)i/2= -п(Ат-2к)(тта?/2. D1.19)
A tensile stress, built up at the tip between the two layers I and II, could
become large enough to create the nucleus of a crack. Since the densities of
pile-up dislocations on the two layers are about the same for long-life fatigue,
except for the sign, the tensile stress axx at x = a and at half the distance
between the two layers becomes
°xx = °fxl + "хУ1 = ЗУ^и Dт - 2к)/Ш, D1.20)
where h is the distance between the two layers. Embryonic cracks can be
formed from this tensile stress as shown in Figs. 41.4 and 41.5. Formation of
the intrusion causes the stress concentration and the intrusion can also be a
crack embryo. The depth of the intrusion equals the total number of accu-
accumulated dislocations which is TV, = TV,, = nAN, multiplied by the Burgers
vector.
The condition of the growth of the crack embryo will be treated from the
viewpoint of energy balance. We assume that when the stored strain energy
due to dislocations accumulated after n cycles becomes equal to the surface
energy, the layers of dislocation dipoles are transformed into a free surface.
The life of the crack initiation nc is now defined as the number of stress cycles
when the following energy condition is satisfied:
U=Uj+Un = 2ncAU= 4aWs, D1.21)

41. Dislocation model for fatigue crack initiation
385
с
Ы Crock Embryo (ill Subsequent Growth
(a) Crack initiation from vacancy dipoles or eitrution
(i) Crack Embryo (iit Subsequent Growth
(b) Crack initiation from intrusion
Fig. 41.4. Two types of crack initiation.
where Ws is the specific fracture energy for a unit area. The right-hand side of
D1.21) is 2aWs for the pile-up shown in Fig. 41.2. Equation D1.21) leads to
nc = 4aWs/(Ar-2k)Ay
= %AWJba{AT-2kf
= 2bWsa3/A{Ayf. D1.22)
The last result,
2 = 2bWsa3/A, D1.23)
(ol Crock Embryo
(b) Subsequent Growth
Fig. 41.5. Grain boundary crack initiated by stacked pile-ups of vacancy dipoles.

386
Chap. 6 Dislocations
1,3, ,2n+P
Fig. 41.6. Stress-strain hysteresis loop.
is similar to the Coffin A954)-Manson A954) law. They have found the life
law for the complete fracture of smooth specimens under high strain cycling.
Later, a similar law was confirmed by Lukas et al. A974) to be valid for
low-strain and long-life fatigue. The life up to the initiation of the crack on a
grain size order is about fifty percent of the total life of fatigue (see Taira et al.
1979). Therefore, the Coffin-Manson relation seems to be valid for the
initiation of cracks in long-life fatigue. The strain energy of dislocations is
accumulated in the same amount in each loading and unloading except the
first loading. The amount of stored energy does not correspond to the total
area of the hysteresis loop, but only to the shaded area shown in Fig. 41.6. The
energy corresponding to the remaining area of the loop is the work dissipated
against the friction stress. Martin A961) derived the Coffin-Manson relation
by regarding the segment of the plastic work associated with work-hardening
as an accumulating damage.
Tomkins A968) also derived the Coffin-Manson law from a different
viewpoint. He assumed that the crack growth per cycle is equal to the product
of plastic strain and plastic zone size. Integrating the growth rate equation
from an initial crack length to a final length, he obtained the Coffin-Manson
law.
The second equation in D1.22) can be written as
AT = 2k + (&AWs/bncf/2a-1/2 D1.24)
which is of the Petch-type equation for constant initial life. The Petch-type
relation has been reported for the long-life fatigue of several metals (see Taira
et al. 1979, Thompson and Backofen 1971, Ebner and Backofen 1959).
The most striking advantage of the Tanaka and Mura model based on the
irreversibility assumption for dislocation motion is that the progress of the

41. Dislocation model for fatigue crack initiation 387
ratcheting of a plastic deformation in the slip band can be calculated in each
loading and unloading. Two neighboring layers, where the motion of disloca-
dislocations with opposite signs takes place, can be regarded as the zone of strain
localization akin to the persistent slip bands found in low-strain fatigue.
Several experiments have been reported on the slip movement in such bands
and most of them indicate a certain irreversibility of plastic deformation.
Keith and Gilman A959) observed in their etch pit study that dislocation
movements were irreversible under cyclic stressing except for a small motion
of dislocation loops in lithium fluoride crystals. More macroscopically, Chars-
ley and Desvaux A969), and Desvaux and Charsley A969) found that a partial
reversal of tensile slip steps occurred during compression, but no total reversal
was seen in single crystals with wavy and planar slip modes.
The dislocation dipole model has been extended to fatigue crack initiations
at inclusions and notches by Tanaka and Mura A982), Mura and Tanaka
A981). The model also has been used for stress corrosion cracking in high
strength steel by Hirose and Mura A984), and Mura and Hirose A984).
Recently, Kato et al. A984) have evaluated the net sum yp of plastic strains
accumulated after и cycles of loading by use of a random stochastic nature of
slips. The possibility distribution function f(yp) becomes the normal distribu-
distribution and the expected value of \yp\ is obtained as {n~Lyp, where n is the cycle
number and Ayp is the applied plastic strain amplitude. Their theory reaches a
result similar to D1.23), but criticizes the energy expression D1.7).

388 Chap. 7 Material properties and related topics
Material properties and
related topics
In this chapter, we demonstrate how inclusion problems are applied to
estimate macroscopic mechanical properties of aggregates, i.e. particle bearing
materials and polycrystalline materials. In these materials, internal stresses
usually develop as a result of plastic deformation which is inherently heteroge-
heterogeneous due to the heterogeneity of the constituent medium. These internal
stresses can be calculated in a straightforward manner by using the methods of
Chapters 2 and 3. If one finds a way to connect these internal stresses with the
applied stress needed to produce the plastic deformation, one can eventually
predict the macroscopic plastic properties.
42. Macroscopic average
We begin with the somewhat general properties of the internal stresses of
inclusions, not explicitly mentioned in the preceding chapters.
Average of internal stresses
UD D2.1)
j
D JD
is rewritten as
dD, D2.2)
since Xj k = Sjk. Integrating by parts and noting aikJc = 0 in D and aJknk = 0

42. Macroscopic average 389
on its boundary \D\, one can see that the above equation is always equal to
zero, that is,
f au AD = f aikXjnk dS - f alktkXj dD = 0. D2.3)
Jd j\d\ jd
Note that this is valid without specifying the elastic constants or the origin of
the internal stress.
Macroscopic strains
Suppose that the eigenstrain ef, is given to an aggregate $2 of inclusions,
producing an internal stress field au and strain field e^. Let us calculate the
average of e,y over the whole body, (е^)о, defined by
<^j)d = (W)j4, dD = (l/V)fD(etJ + e*) dD, D2.4)
where V is the volume of the body and etj the elastic strain. If the elastic
constants are uniform, D2.4) is rewritten as
<<,j>D = (VV)CrJk\(okl dD + A/V) f ?*¦ AD, D2.5)
which becomes
N,dD D2.6)
from D2.3). Since ef, is defined only in fi, the integral domain in D2.6) can be
changed to ?2. By defining the volume fraction / of the inclusions as ti/V,
D2.6) is written
<*,,>W<e?>=Mr D2.7)
The last result applies when c*7 is uniform in ?2. Expression D2.6) can be used
to describe macroscopic strains induced by dislocation motions.
When the inclusions have elastic moduli different from those of the matrix,
the inhomogeneous inclusions are simulated by the equivalent inclusions

390 Chap. 7 Material properties and related topics
defined in Section 22. That is, when the eigenstrain of the inclusion is «fy, we
have
°,l = C,lkl{4i-ttT) = CrjuD,-<li) inJ2, D2.8)
where
еГТ=<Ь + *Ь, D2.9)
<u = SUmni*m*n, D2.10)
and C*jkl is the elastic moduli of the inhomogeneous inclusions. Then D2.6)
and D2.7) are written, respectively, as
/o( ;) A/V)fj** &D D2.11)
and
<<„>/> =/«?,*• D2.12)
Hence, the macroscopic strain is directly related to the eigenstrain of the
equivalent inclusion.
In the above procedure, it is assumed that the stress due to a particular
inclusion is not disturbed by other inclusions. This approximation is permitted
when / «: 1. However, it is a poor approximation when / is large.
Tanaka-Mori's theorem
For simplicity, let us consider a homogeneous body containing an inclusion Q
not necessarily ellipsoidal (see Fig. 42.1). At this stage of discussion, the shape
of the inclusion is not specified. Give to the inclusion a uniform eigenstrain
e*. Let the internal stress due to e* be a™, implying that a," is calculated by
assuming the body to be infinitely extended, o™ outside $2 is safely given by
a~(x) = -CijkljQCpqmif.*mnGkp,ql(x- x') их' D2.13)
from C.25), where e* „ = 0 for points exterior to 12. Let us integrate a* over

42. Macroscopic average
391
Fig. 42 1. Inclusion Q with an arbitrary shape in an infinite domain.
the domain V2 — Vx as shown in Fig. 42.1. V2 and Ул are ellipsoidal and
V2 d Vx о Q.
o,7 dx=-Cijklf dxjCpqmnt*mnGkpql{x-x')dx'
D2.14)
When we change the order of integration in D2.14),
f a» dx = - / dx' CIJklf Cpqmn<i*mnGkp,ql{x - x') dx
+ f dx'lcijklf Cpqmn€*mnGk ,(x-x')dx
Since
D2.15)
- / Cpqmne*mnGkPtql{x - x') dx
gives the total distortion at x' when the uniform eigenstrain is defined in V2
and we have assumed V2 to be an ellipsoid, this distortion is independent of jc'
and is determined by the shape of V2, as long as x' e V2. This condition is
satisfied since x' e ?2 с V2. A similar remark applies to the second term in
D2.15). Thus, D2.15) is rewritten as
= QCljkl[Sklmn(V2)e*mn -
D2.16)

392 Chap 7 Material properties and related topics
[see A1.1) and A1.15)], where i2 is the volume of the inclusion, and Sklmn(V2)
and Sklmn{Vx) are calculated by A7.14) as
Sklmn
D2.17)
Equation D2.16) leads to interesting conclusions: The volume integral of o™
over V2 - V1 is proportional to Й; it is independent of the absolute position
and size of V2 and Vx as long as V2 э Vx; it depends only on the shape of V2
and Fj; and it vanishes when V2 and Vx are similar in shape and have the same
orientation, that is, when ax/bx = a2/b2 = a3/b3, and
V2: x\/a\ + x\/a\ + x\/a\ < 1,
V,: (Xl - AJ/b\ + (x2 - x°2JM + (x3 - x°3J/bj < 1.
The last conclusion D2.16) is valid even when the eigenstrain in J2 is
non-uniform and the elastic moduli of Q differ from those of the matrix.
When J2 is an ellipsoidal inclusion we can have the limiting situation,
V-y -> J2. Then
, V2-> ill. y4i..iy)
If V2 and п are similar in shape, D2.19) vanishes,
( o°? dx = 0, V2 э 0. D2.20)
•/F2-fi
Similarly, for the corresponding strain,
j e?} dx = 0, V2 з fi. D2.20.1)
The above properties of the integration in the domain around an ellipsoidal
inclusion have been obtained by Tanaka and Mori A972). Brown and Stobbs
A971) have noted D2.20) by using explicit form of a" due to a spherical
inclusion in an isotropic medium. It should be noted that D2.20) is valid for
an inhomogeneous inclusion with arbitrary and non-uniform eigenstrains:

42. Macroscopic average 393
Fig 42 2. Domains ii, Vl and V2 have the same shape and orientation.
D2.15) becomes zero even if t*mn = ??„(*') when V2 and Vx have the same
ellipsoidal shape and orientation (see Fig. 42.2).
Image stresses
Let us consider a finite body D containing an ellipsoidal inclusion ?2 with
eigenstrain efr In order to utilize the property D2.20), we assume that D is
similar to ?2 in terms of its orientation and shape. This assumption does not
restrict the problem when Q is very small compared with D.
Let us find the internal stress atJ due to Q. In the first approximation, atj is
taken as a* defined previously. The integration of a~ over the whole body
can be written as
f o% dD = f o°° UD+ f < dD. D2.21)
Because of D2.20), the second term vanishes; thus
/"<dZ>= [ofidD. D2.22)
JD Ji2
Equation D2.22) shows that a™ does not possess the property D2.3) of
internal stresses in a finite body and is based on the fact that a" does not
satisfy the traction free condition o^rij = 0 on \D\. Thus, one must add the
correction term a'j to o°° to obtain the exact solution for the internal stress
a,.y = a,7 + <, D2.23)
where a,jj = 0 in D and а,^иу + o°°nj = 0 on \D\. a,'j is called the image
stress. The determination of this image stress is generally a difficult problem.

394 Chap. 7 Material properties and related topics
However, one can estimate its average value from D2.3), D2.23), and D2.22).
This average is
<",'>/> = (W)JV, dZ> = - (l/K)JU" dZ) = - ?/V)jo% dD, D2.24)
where V is the volume of D.
When the eigenstrain t* is uniform in J2, D2.24) becomes
(afj)D= -(Q/V)o$(Q)= -{Q/V)Cijkl{Sklmn(O)e*mn-eti}- D2.25)
(a,IJ)D is small compared to the magnitude of a". Nevertheless, when afj and
a" are integrated over the whole body, they attain values of the same order.
We can obtain a result similar to D2.25) for strain. Let
?,., = ??+??,., D2.26)
where cf • is the image strain due to the presence of the free surface. Equation
D2.20.1) leads to
f^ dD = (a/V)Sljkl(a)€t,, D2.27)
while from D2.6) we have
<<u)d = <<">/> + K)d = (V/Vhty D2.28)
Therefore, combimng these results, we have
D2.29)
which can be obtained directly from D2.25) by inspection.
The importance of the image stress and strain has been discussed by
Eshelby A956) in connection with macroscopic deformations and average
changes in the lattice constants.
Random distribution of inclusions—Mori and Tanaka's theory
Seldom is only a single precipitate formed in a material. For example, when
the volume fraction of precipitates is 0.001 and their size is 100 A, some 10
precipitates are present in a unit volume. Even if the stress field due to an

42. Macroscopic average 395
individual precipitate were known, including the image stress term, it would
take many years to add up the stresses due to all these precipitates. Moreover,
the positions of individual precipitates are generally not known.
However, the average stress in the matrix and in the precipitates (inclu-
(inclusions) as well as the elastic energy are easily calculated when the precipitates
are randomly distributed. Sometimes all these quantities are needed to discuss
the macroscopic properties.
For simplicity, homogeneous ellipsoidal inclusions, with uniform eigen-
strain «*, are treated first. Let us formally define the average stresses in the
matrix as {otj)M and those in the inclusions as (o^}/. Equation D2.3) can
then be expressed as
/<a,.,>/+(l-/)<a,7>M = 0, D2.30)
where / is the volume fraction of the inclusions. Now let us introduce a new
single inclusion. The number of inclusions is so large that this addition does
not affect /. Since random distribution of the inclusions is assumed, we will
insert the new inclusion randomly in the matrix. Then, within the inclusion (or
within any inclusion) the stress becomes
a,7 = a~ + <a,,>M, D2.31)
where a,°° is the stress calculated for a single inclusion present in an infinite
medium:
^ = Cijkl(Sklmnt*mn-4,), D2.32)
and (ojj)m is the stress already present in the matrix. Since the new inclusion
can be inserted at any place in the matrix, the average of atJ must be equal to
(ои)[. The average of D2.31) over all the positions available for inclusions is
(<ylJ)I = o^ + (oij)M. D2.33)
Substituting D2.33) into D2.30), the following equations are obtained:
<°v)i«=-fav> D2-34)
<%), = °U-f°V- D2-35)
It should be noted that —/a" in the above equations is the average of the sum
of o™ and afj caused by all the inclusions.

396 Chap. 7 Material properties and related topics
The elastic strain energy per unit volume of the body is written as
w* = -И/Ч dr>/v= -bb<av>if
= -1/A-/)<«* D2.36)
from D2.35) and B5.2) or A3.3). It is extremely useful that the average stress
and the elastic energy can be written in such simple forms when the inclusions
are randomly distributed. This fact has been found by Mori and Tanaka
A973). Brown A973) has considered the same problem and has obtained a
slightly different result.
A parallel discussion can be given in the situation when the eigenstrain is
not uniform in an inclusion. In such a case, D2.33) can be written as
<*,7>/=«>/+<°,7>M D2-37)
with
<<>,== (I/O) jTo^dD, D2.37.1)
where o™ is the stress for a single inclusion present in an infinite medium and
?2 is the volume of the inclusion. Then, D2.30) and D2.37) lead to
(%)м =-/«>/, D2.38)
<°0>/= (!-/)«>/¦ D2.38.1)
From D2.31) the stress inside a representative inclusion can be written as
°-7 = < -/«>/• D2.39)
When the inclusions are inhomogeneous with eigenstrains efj, the treatment
may be slightly modified. Let us assume, for simplicity, that e^ is uniform in
the inhomogeneous inclusions. The equation D2.30) is still valid. The average
stress in the matrix can be written as
(%)м=Сик1е°к1, D2.40)
where e°kl is to be determined. Let us introduce a new single inhomogeneous

42. Macroscopic average 397
inclusion into the matrix, as before. Then, the stress in this inclusion can be
determined by the equivalent inclusion method B2.13),
°tj = ClJkl{el,+ Sklmne*m*n - 4f) = C*kl(e°kl+Sklmne*m*n - cf,), D2.41)
where C*kl are the elastic moduli of the inclusions. The stresses atJ in D2.41)
can be assumed to be (a,-y-O. Then, D2.30) leads to
fC,jkl{e°kl + Sklmne*m*n ~ 4Г) + A -f)CIJkle°kl = 0 D2.42)
or
fCijkl(Sklmnt*m*n - 4Г) + CiJkle°kl = 0. D2.43)
Equations D2.41) and D2.43) are necessary and sufficient to determine 12
unknowns e°j and ?**.
If /•« 1, the following approximation is admissible. Setting (.** = e? + efj,
(tf =)CiJkl(SklmitZn - ??,') = Crjkl{Sklmn^n - 4,) D2.44)
and
(а;^)Сик,(е°к1+ Sklmnc*mn - e*kl) = C*Jkl{e°kl+ Sklmnt*mn), D2.45)
where e*' and tfj are the respective equivalent eigenstrains for ef; and e,°.
It is easy to solve D2.44) for ef,' under given cfr We assume that efy- ¦« t*'ir
Then, D2.43) leads to
<°и)м=Сик1е0к1*>-Го%. D2.46)
Substituting e°j obtained from D2.46) into D2.45), we can derive approximate
values of tfj. Then, D2.41) can be written as
<°,-,>/=°" + ffo- D2-47)
It is apparent that when /<tc 1, D2.38) is also a good approximation for the
situation where e? is not uniform in an inclusion. The discussion of this
problem is similar to that developed above.

398 Chap. 7 Material properties and related topics
43. Work-hardening of dispersion hardened alloys
Metallurgists invented a dispersion strengthened alloy whose matrix is a
plastically soft and ductile material and throughout which particles of plasti-
plastically strong second phases are dispersed. Sometimes these particles are called
nonshearable, implying that the particles cannot be cut by the crystal disloca-
dislocations (glide dislocations). These particles thus hinder the dislocation motion
and thereby raise the yield stress. Macroscopic yielding in a dispersion
strengthened alloy is explained by the Orowan mechanism (see Ashby 1966);
dislocations bowing out between particles. The critical stress for this bowing
out can be calculated by using the results C6.15) or C6.25). For recent
developments of this problem, the reader is referred to Bacon et al. A973) and
Scattergood and Bacon A975). When dislocations bow out, dislocation loops,
called Orowan loops, are left around the particles. The internal stresses from
these Orowan loops affect the motion of the dislocations. Macroscopically,
this effect is manifested by work-hardening. One approach used to estimate
work-hardening is the balancing of forces acting on the glide dislocations
bowing out between two neighboring particles, where one force comes from
the applied stress, one from the glide dislocation itself, and one from the
internal stresses caused by the Orowan loops. An approximate method using
this assumption has been employed by (Hart 1972). However, this type of
approach depends critically on the model adopted, the position of the loops
around a particle, as well as the number of the particles considered. A
particular loop may hinder or assist the motion of a dislocation, depending on
the relative position of the loop and the dislocation. It is, therefore, desirable
to take another approach, with which the estimation of work-hardening can be
made independently of the choice of a detailed model. The following sections
are devoted to this purpose and are mainly applications of the analysis of the
elastic fields produced by an ellipsoidal inclusion.
Work-hardening in simple shear
Let us consider the situation where the matrix deforms plastically but the
inclusions do not. For simplicity, the shape of the inclusions is assumed to be
spherical and the plastic deformation in the matrix is assumed to be uniform.
As an example, the plastic deformation is assumed to occur in simple shear;
the slip plane is thus perpendicular to the x3-direction, while the slip direction
is parallel to the .^-direction. Let the plastic strain in the matrix be t%x = \yp-
(yp is the glide strain in the slip system.) Superimpose efj = — \yp uniformly
throughout the specimen. Since the uniform plastic strain over the entire body
does not produce internal stress, the internal stress induced by the prescribed

43. Work-hardening of dispersion hardened alloys 399
plastic deformation in the matrix is calculated by considering the situation
where
efj = 0, in the matrix,
(¦ъ\= -\yp, in the inclusio
From B2.13a) and B2.19), the stress in an inclusion is given by
(¦ъ\= -\yp, in the inclusions.
pg D3.2)
when the inclusion is isolated in an infinite matrix. Here /t and ju* are the
shear moduli of the matrix and the inclusion, respectively,
D3.3)
and
v)}, D3.4)
where v is the Poisson ratio of the matrix. From D3.1), D3.2) and B5.3), the
elastic strain energy due to an inclusion is calculated as
W* = \V0nn*(l-2S)yp2/g, D3.5)
where Vo is the volume of the inclusion. When many inclusions coexist with a
volume fraction /, the elastic strain energy per unit volume can be approxi-
approximated by
W* = \fw*{l-2S)^/g, D3.6)
provided / is much smaller than 1. The otherwise uniform applied stress,
a3Oj = a0, is disturbed by the inhomogeneity in the elastic constant of the
inclusions. The applied stress in the inclusion is calculated as
a31 = 0o{l + Oi--/i)(l-2S)/g} D3.7)
from B2.7). Thus, the change in the external potential per unit volume V* is
given by B5.40),
V* = -a°yp +fa°{l + (M* - M)(l - 2S)/g}yp. D3.8)

400
Chap. 7 Material properties and related topics
Ур/g
Fig 43.1. Theoretical lines of linear work hardening due to particles with volume fraction /
The Gibbs free energy B5.41) per unit volume of the specimen, AW, is the
sum of W* and V*,
D3.9)
The specimen is in stable equilibrium when dAW/dyp = 0 and d2AW/dyp > 0.
That is, the stable state is achieved by
°°{A -/) -/(M* - /0A - 2S)/g) =/№*A - 2S)yp/g.
Since /<к 1 is assumed, D3.10) can be written as
ff°=/Wi*(l-2S)Y./g,
D3.10)
D3.11)
which represents the equation relating the applied stress and the plastic strain
(the constitutive equation) as is schematically shown by the dotted line in Fig.
43.1.
Since a finite stress is needed to start the plastic deformation, one has to
raise the curve to the solid line,
° = oy+fw*(l-2S)yp/g.
D3.12)
The physical_reason for this is as follows: When the increase in yp is Syp, the
increase in W* is /juju*A - 2S)yp8yp/M, and the work done by the applied
stress (negative of the external potential increase, SU) is o°8yp. At the same
time, the increase in yp in the matrix accompanies the energy dissipation

43. Work-hardening of dispersion hardened alloys 401
о
b
Cu-SiO,
f • 0.0052
0 2 4 6 8 10
Ур (Ю-')
Fig. 43.2 The solid line is the experimentally observed stress-strain curve and the dotted line is
calculated from D3 11) (after Tokushige 1976)
SD* * oy8yp. The energy conservation law, 8U= SW* + SD*, yields the solid
line in Fig. 43.2, which indicates linear work-hardening after yielding. Yielding
by the Orowan mechanism is typical of energy dissipation. Thus, ay can be
identified as the critical stress for the operation of the Orowan mechanism.
The above method of treating the plastic deformation of dispersion-
strengthened alloys appeared in the paper by Tanaka and Mori A970).
Experimentally, linear work-hardening is usually observed in a dispersion-
hardened alloy when the strain is small and the deformation occurs at a low
temperature, where the diffusion effect can be ignored. An example (Tokushige,
1976) is given in Fig. 43.2, in which the solid line is the experimentally
observed stress-strain curve of a Cu-SiO2 single crystal and the dotted line is
calculated from D3.11). The SiO2 particle is known to have a spherical shape
in this alloy. Thus, S = D - 5e)/{15(l - v)}. Here /= 0.0052, ju = 4.61 X 1010
N/m2, ц* = 3.13 X 1010 N/m2 and v = 0.33. The calculation accounts for 65%
of the observed hardening.
It is easy to extend the above calculation to other shapes of inclusions as
well as other modes of plastic deformation such as uniaxial deformation as
shown already by B5.58) (Tanaka and Mori, 1970). The elastic anisotropy of
the matrix and the inclusions can also be taken into account (Lin et al., 1973).
The anisotropy in plastic deformation induced by the asymmetrical arrange-
arrangement of inclusions in a composite material has been discussed with an
approach similar to the above method (Tanaka et al., 1973).
Brown and Stobbs A971), and Brown and Clarke A975) have taken a
different approach to derive the hardening rate, a method essentially identical
to D3.11), although their paper contains some misleading interpretations.
These have been corrected later by Mori and Tanaka A973). From D2.46) and
D3.2), the average internal stress in the matrix is given as
l-2S)yp/g, D3.13)

402 Chap. 7 Material properties and related topics
after the matrix has been plastically deformed by yp. Thus, to continue the
plastic deformation, one has to raise the applied stress by
Vi=-<*3i>m D3.14)
to overcome the internal stress. Note that D3.14) is identical to D3.11). Both
the stress fields due to the inclusions in the infinite matrix and the image
stresses due to the free surface effect contribute to (ст31)м. — (о^)м is
sometimes called the back stress. It is clear that (о^)м hinders the progress of
plastic deformation, and aids the plastic flow in the reverse direction. Of
course, one can see this from the preceding energy consideration. This prop-
property should be manifested in the Bauschinger effect. Conversely, the existence
and magnitude of the back stress can be examined from the measurement of
the Bauschinger effect. Some studies have been done along this line (Atkinson
et al., 1974, Gould et al., 1974, and Mori and Narita, 1975). The magnitude of
the back stress, — (o3l)M, determined from the difference of the flow curves
between forward and reverse strainings, is well accounted for by D3.13) when
strain is small.
Let us examine D3.10) and D3.11) further. These equations account for the
interaction among inclusions and for the free surface effect. The procedure
mentioned in the preceding section, D2.40) through D2.47), can be used to
improve these approximations. Then, the average internal stress in an inclu-
inclusion is evaluated as
<°3i>/ = №*A - 2S)yp{A -/) -/(/x* - M)(l - 2S)/g) g, D3.15)
and W* now is
W* = i/^*(l - 2S)Y/{A -/) -f(ii* - M)(l - 2S)/g)/g. D3.16)
Writing AW with D3.8) and D3.16), we obtain D3.11) instead of D3.10), as an
equation satisfying the equilibrium condition.
Dislocations around an inclusion
Here we discuss the dislocation distribution around an inclusion when е%г = \yp
is uniform in the matrix and e^ = 0 in the inclusion. For simplicity, assume
(Рз1=УР in the matrix,
\fifj-0 in the inclusions.

43. Work-hardening of dispersion hardened alloys
403
(а) (Ы
Fig. 43.3. Surface dislocations around a particle.
Since there is a jump at fit at the matrix-inclusion interface, the surface
dislocations ahi defined by C7.16) exist at the interface. Let n be the unit
normal vector at the interface directed toward the matrix. Then,
«11= -
D3.18)
All other components are zero. This means that the infinitesimal glide loops,
which have the Burgers vectors parallel to the loop planes, are present at the
interface (see Fig. 37.2 and Fig. 43.3(a)). The total Burgers vector of the
surface dislocations crossing the unit length of the line element perpendicular
to v and lying on the interface, becomes
В\ = «ll^l + «21  = УрA ~ nl) / .
since
D3.19)

= ~п2/{п\
) /
D3.20)
This result D3.19) implies that the density of the dotted lines in Fig. 43.3(a) is
a constant equal to yp along the x3-axis.
With the dislocation distribution defined previously, another physical
explanation can be given for the work-hardening derived in this section.
Physically, the plastic deformation is caused by the motion of the crystal

404 Chap. 7 Material properties and related topics
dislocations with a finite Burgers vector. Thus, the situation described by
D3.17) does not generally occur literally, say, at the atomic dimension. All that
D3.17) implies in the case of crystal plasticity is that slip occurs randomly, and
macroscopically uniformly. Here, "macroscopic" may be defined in such a
manner that uniformity is discussed in comparison with the inclusion dimen-
dimension. For example, D3.19) shows that the number и of the (crystal) Orowan
loops surrounding a particle is, on the average, given by
n = 2ayp/b, D3.21)
and that these loops are randomly distributed around the particle. Here, b is
the magnitude of the crystal dislocations Burgers vector responsible for slip,
and a is the radius of the spherical inclusion. This situation is schematically
shown in Fig. 43.3(b). When the inclusions are randomly distributed, it is
apparent that the position of the Orowan loops around a particular inclusion
is also random. Let N dx3 be the number of the Orowan loops per unit
volume, situated between x3 and x3 + dx3 with respect to the center of an
inclusion (see Fig. 43.3). N is called the Orowan loop density. Then, uniform-
uniformity of the plastic deformation in the matrix means that N is constant related to
Y, by
f
J —
Ndx3 = nN D3.22)
or
yp = Nb/Np, D3.23)
where Np is the number of inclusions per unit volume.
Now let us examine the stress field due to these Orowan loops. Consider the
average of the internal stress in the matrix. Since there are many loops, this
average remains almost unchanged when a loop is transferred from one
particle to another. We repeat this until some particles are uniformly covered
by the loops (other particles may not have any loop), then, divide these loops
into the loops with an infinitesimal Burgers vector and redistribute these
infinitesimal loops to all the particles uniformly. It is clear that, in the final
stage, the infinitesimal loops are distributed uniformly as the surface disloca-
dislocations described in D3.18) or D3.21), and that the average internal stress in the
matrix is the same as that in the original distribution of the Orowan loops with
a finite Burgers vector. In short, the work-hardening discussed in this section
is that brought by the average internal stress due to the Orowan loops. For the
discrete dislocation approach, it is difficult to conduct a quantitative calcula-

43. Work-hardening of dispersion hardened alloys 405
tion. Such difficulty can be avoided by using the stress field of an inclusion, as
has been suggested by Mori and Tokushige A977).
When the situation arises that the Orowan loop distribution density N is
not constant but is a function of x3, the average internal stress in the matrix
may still be calculated by the above procedure. The average internal stress in
the matrix can be reproduced by giving the eigenstrain
^=-N(x,)b/2Np D3.24)
to every particle. Then, the solutions of Sections 19 and 20 can be used
together with D2.37.1). Such a situation will be discussed in the next section.
Uniformity of plastic deformation
One of the reasons that the straightforward evaluation of work-hardening in
dispersion strengthened materials is possible is the assumption of uniform
plastic deformation throughout the matrix. In the context of the approach
taken in the present section, this assumption is justified, as shown below.
For simplicity, let us assume /¦« 1 and that the plastic strain in the matrix
efj varies only along the x3-direction, that is,
<?i = H,(*3). D3-25)
Neglecting the interaction among the inclusions and the disturbance of the
applied stress a31 = a0 due to the inhomogeneity of the inclusions, one can
write the Gibbs free energy per unit volume of the specimen as
dx3- f o°yp(x3)dx3 D3.26)
with
A=p.*(l-2S)/g. D3.27)
Here, the energy dissipation is not considered and Z>3 indicates the integral
domain with respect to x3. The first term in D3.26) represents the elastic strain
energy (see D3.6)). The second term arises from the external potential. It is
seen that D3.26) takes a minimum when
D3.28)

406 Chap. 7 Material properties and related topics
That is, the stable state is achieved when the specimen is uniformly covered by
the plastic domain in which the matrix deforms plastically with constant у
determined by a0 in D3.28). As mentioned before, uniform slip cannot occur
in a literal sense, since crystalline materials deform plastically by movement of
crystal dislocations. Therefore, equation D3.28) should be interpreted in such
a manner that the slip lines, each of which is produced by a single dislocation
movement, cover the specimen uniformly, i.e., the slip line spacing becomes as
small as possible. With this conclusion and the consideration of the dislocation
configuration around the inclusion discussed previously, the procedure to
evaluate the work-hardening is justified. The above procedure and conclusions
are identical to those given by Mori and Mura A976).
44. Diffusional relaxation of internal and external stresses
Let us consider the elastic deformation of an infinite body due to a^. When
the body contains an inhomogeneity, the stress field is disturbed. When the
inhomogeneous particle is harder than the matrix, the interface boundary of
the matrix (dotted line) and the particle (solid line) deform as shown in Fig.
44.1, if the two domains deform freely without any constraint from the
surroundings. The internal stress (stress disturbance) can be interpreted as the
eigenstress caused by the eigenstrain corresponding to the misfit (the dif-
difference between the dotted domain and solid domain in Fig. 44.1). If interfa-
cial diffusion takes place on the particle surface, mass must be transported
from region I to region V. The misfit would therefore disappear, and the
internal stress and elastic energy would decrease; this is diffusional relaxation.
Similarly, diffusional relaxation takes place when the matrix is subjected to
plastic deformation but the inclusion is not.
In this section, some experimental evidence of diffusional relaxation will be
mentioned. Next, the relaxation rate will be discussed and the creep deforma-
-y\
Fig. 44.1 When an inhomogeneous particle is harder than a matrix, the interface boundary of the
matrix (dotted line) and the particle (solid line) deform as shown here by o30, if no constraint
exists from the surroundings (%).

44. Diffusional relaxation of internal and external stresses 407
tion will be considered as a balance between the accumulation and the
relaxation of the internal stress. The diffusional relaxation of the external
stress will be also discussed.
Relaxation of the internal stress in a plastically deformed dispersion strengthened
alloy
Suppose that plastic deformation efj = \yp occurs only in the matrix. The
internal stress in this state is caused by the eigenstrain given in D3.1), which
can be understood from Fig. 44.1. If a complete relaxation occurs by a
diffusional process which transports mass from region I to region V in Fig.
44.1, then the eigenstrain in the inclusion would be reduced by
Mi = hP D4.1)
and the resulting eigenstrain in the inclusion would become zero. In that case,
work-hardening is lost by the amount given by D3.11); the flow stress
decreases correspondingly. Figure 44.2 shows the flow stress decrease of
Cu-SiO2 crystals (/= 0.0052) work-hardened at 77°K for 20 min. annealing at
473° К (Mori and Tokushige 1977). The dotted line is drawn according to
D3.11). Except for the high strain region, the agreement between the predict-
prediction and the observation is good. Metallurgically, the deviation at the high
strain region indicates that some relaxation processes which do not need
diffusion have occurred.
The diffusional relaxation mentioned in this section can be further checked.
The additional eigenstrain, D4.1), should be observed by the macroscopic
deformation. By using D2.8), D2.12) and B2.19), the macroscopic strain
increment (e3l)D due to Ae^ in the inclusions can be calculated as
<«3i>z>=/«5i* D4.2)
with
etf=/iM€&/g, D4.3)
where g is defined in D3.3). From D3.11) and D4.1) through D4.3), the loss of
the work hardening, Aa°, is related to the macroscopic strain observed in
relaxation annealing. It becomes
Ac0 = 2^A -2S)(t31)D. D4.4)

408
Chap. 7 Material properties and related topics
Shear Prttrroln (%)
Fig. 44.2. The flow stress decreases by diffusion after prestraming (after Mon and Tokushige
1977).
This has been tested in Cu-SiO2 crystals as shown in Fig. 44.3 (Mori and
Narita 1975). The specimens were prestrained and annealed. The change in
flow stress, Aa°, and the change in specimen length (corresponding to (e3i)D)
due to annealing were measured. We assume that the annealing causes Ae^ in
D4.3). The theoretical values from D4.4) and the corresponding measured
values are plotted by the filled circles and the triangles, respectively. These are
in good agreement. The open circles represent the change in the back stress,
- (on)M, due to annealing. These changes were determined by Bauschinger
tests before and after annealing. —(an)M is calculated as half of the dif-
difference of the flow stresses between forward and reverse straining. The
changes in — (an)M also compare well with Aa°. The saturation tendency of
Aa° with prestraining is consistent with Fig. 44.2. Since the specimens were
considered to have annealed fully, the change in back stress by annealing is
equal to the back stress present before annealing. The saturation of the back
stresses at high prestrains in Fig. 44.3 agrees with an observation by Atkinson
et al. A974).
Diffusional relaxation process, climb rate of an Orowan loop
There are tree types of diffusion: volume, interfacial and pipe diffusion. The
last one is caused by mass transport along the core of a dislocation, where
diffusion is fast with a low activation energy because of the severely disturbed
atomic arrangement. In this section we employ the pipe diffusion as a cause of
climb of the Orowan loops. The interfacial diffusion will be examined in
connection with stress relaxation in later sections.
The climb process of the Orowan loop is explained by Fig. 44.4. If atoms
flow from the left-hand side of an Orowan loop to the right-hand side through

44. Diffusional relaxation of internal and external stresses
409
Fig 44.3 The decrease in the back stress due to the recovery against the prestrain (after Mori and
Nanta 1975). The axis of abscissa is shear prestrain (%)
its own pipe, the Orowan loop will climb up. Let us calculate the number of
atoms, 8N, diffusing when an Orowan loop with the Burgers vector b climbs
from x3 to хъ + Sx3. The particles are assumed to be spherical with radius a.
Since the thickness of an extra plane defined by the edge component of a
dislocation is expressed by the magnitude of vXb = (b sin в) (see Fig. 44.5),
where v is the unit vector along the dislocation, SN is calculated as
8N = A/0)8*3 f'b sin в(а2 - xjf/2 dO = 2b(a2 - xjI/28x3/i2, D4.5)
where Q is the atomic volume. The elastic energy of the Orowan loop W* is
Fig. 44.4. The climb process of the Orowan loop.

410
Chap. 7 Material properties and related topics
Fig. 44.5. Explanation for equation D4.5).
given by
D4.6)
(see, for example, Barnett 1976). Here, a is a constant of the order of unity
and ju is the shear modulus. Thus, the change in the elastic energy 8W* is
SW* = -2<пщЪ2х38х3/{а2 - xj)
1/2
D4.7)
when the loop climbs up by 8x3. The diffusion distance of the atoms, /, is
approximated to be a constant given by
1=ш{а2-х\)Х/1 D4.8)
which is the average over the possible travelling distance. One can define the
force F acting on an atom as -(8W*/18N). Thus,
F=aixb?2x3(a2-x2) V\
The Einstein relation gives the drift velocity v of an atom as
v =
j)~3/2
а2 - xj)~3/2Dp/kT,
D4.9)
D4.10)
where Dp is the pipe diffusion constant along the dislocation, к is the
Boltzmann constant, and T is the temperature. Let us define by h2 the cross
section of the dislocation pipe. During the time interval 8t, atoms 8N =
2h2v8t/Q flow and consequently the Orowan loop climbs by 8x3 related to
8N by D4.5). Thus, the climb rate, F= dx3/dt, of the Orowan loops is given
by
V=^=Ax3(a2-xiy2 D4.11)

44 Diffusional relaxation of internal and external stresses 411
with
A=wuh2Dp/kT. D4.12)
The above treatment of the climb rate of an Orowan loop is developed by
Mori and Tokushige A977). The treatment can be easily extended to the case
when the inclusions are ellipsoidal (see Okabe and Mori 1979). Hazzledine and
Hirsch A974) have taken a different approach to analyze the climb process of
the Orowan loops. Sato et al. A984) have calculated the change of elastic
distortion caused by annealing of Orowan loop and compared with experi-
experiments.
Suppose that the matrix deforms with a strain rate of Цх = \yp. If slip
occurs uniformly, from D2.23) the Orowan loop density distribution N
increases by ypNp/b per unit time, Np being the number of inclusions per unit
volume. At the same time, the loop density decreases with a rate of
-d(NV)/dx3. Thus, in total, one has a differential equation,
When yp = 0,
? = -« D4.14)
at OX3
corresponds to annealing. It can be shown that the solution of D4.14) with the
initial condition of N(x3, 0) = NQ is
N(x3,t)=N0Z{° Хз) , D4.15)
x3(a2-z2)
where z is the root of
At = a*\og{x3/z)-\(x2-z2){Aa2-x2-z2). D4.16)
Mori and Mura A978) have solved D4.16) numerically to obtain JV(jc3, t)
which may be approximated by N = N0(m0 + m2xf/a2) if proper constants
m0 and m2 are chosen. No is related to the prestrain Yp which has induced
work-hardening.
From D3.24), one can express the eigenstrain «fj needed to calculate the
back stress —(o3i)M in a polynomial form. Thus, by using the result of

412 Chap. 7 Material properties and related topics
Section 19 and B2.13), one can calculate a^, by which - (o31)M is obtained
from D2.38) by assuming /«I. Mori and Mura A978) have numerically
calculated the back stress decrease of a work-hardened Cu-SiO2 alloy and
compared it to the experimentally observed softening on annealing. Although
the calculation reasonably accounts for the observation, it predicts the temper-
temperature range at which significant softening would occur to be narrower than the
observed range. This has been corrected by Mori and Osawa A979) who have
taken the distribution of the particle size in an actual alloy into consideration.
It is to be noted that the climb rate of the loop and thus the loop distribution
brought about the relaxation are sensitive to the particle size.
Recovery creep of a dispersion strengthened alloy
If the Orowan loop density distribution is in balance between loss due to
climb, and gain due to the accumulation by plastic deformation, one would
expect stationary creep. The Orowan loop density distribution corresponding
to a stationary creep rate yp is obtained from
since bN/dt = 0 in D4.13). Since V is given by D4.11), it is clear that the
following equation satisfies D4.17):
N = yp(Np/b){a2-x2J/A D4.18)
with A given by D4.12). From D3.24) and D4.18), the eigenstrain given to a
particle for calculation of (ои)м is written as
ep31=-yp(a2-x23J/2A D4.19)
which is in the form of A9.1). Thus, using the results of Section 19 (Asaro and
Barnett 1975, Mura and Kinoshita 1978) and the equivalency condition
developed in Section 22, one finds the stress in the inclusion as
o? = ypa4 (Bo + B^xpxq/a2 + Brsxpxqxrxs/a4)/2A, D4.20)
when the inclusion is in an infinite matrix. The coefficients Bo, B$q and В?я"
are all computable and expressed in terms of the elastic constants of the

44. Diffusional relaxation of internal and external stresses
413
matrix and the inclusion. Thus, by D2.38) and D2.37.1) one obtains the
average of the internal stress in the matrix as
D4.21)
where Ao, A2 and A4 are given by Okabe et al. A980). Therefore, the applied
stress a3Oj = a0 needed to maintain stationary creep deformation is given by
nfypa4(A0-2A2
D4.22)
where aD is the flow stress of the matrix (in absence of any particle).
The experimental observations relevant to D4.22), such as the existence of
the stationary creep and the linear dependence of y^ono0 have recently been
made (Mori and Osawa 1979, Okabe and Mori 1979, and Okabe et al. 1980).
Figure 44.6 shows the observed stationary creep rate plotted against the
applied stress in a Cu-SiO2 alloy (single crystals, /=0.005 [Okabe et al.
1980]). In a certain range of a0, yp is linearly related to a0, in accordance with
D4.22). One can also examine the particle size dependence of creep rates. The
4.0
о
— jo
о
or
a.
I»
о
с
о
2 i.o
Large Porticle Specimen
•KI3K *J93K
x Temperature Change
Small Particle Specimen
о 293 К
Cu-S
293K
55 nm
20
25 30 35 40
Applied Shear Stress (MPa)
Fig 44.6 Theoretical lines and experimental values of the stationary creep with parameters of
particle sizes and testing temperatures.

414 Chap. 7 Material properties and related topics
ratio of the slope of the creep rate for the specimen with a = 45 nm to that for
the specimen with a = 55 nm is obtained as 2.17, which is in excellent
agreement with E5L/D5L = 2.23, as required by D4.22). Finally, let us
compare the calculation with the absolute magnitude of the slope in Fig. 44.6.
By conducting numerical calculations, one obtains
a
0 = a° + 0.5838/i/vpa4/2A. D4.23)
Here, ц is the shear modulus of the matrix (Cu, 4.61 X 1010 N/m2). In the
calculation, Poisson's ratio of Cu has been taken to be 0.33, and the shear
modulus and Poisson's ratio of SiO2, 3.13 X 1010 N/m2 and 0.18, respectively.
By assuming a = \, Я = 11.8 X 100 m2, h2 = b2, b = 2.56 X 100 m, Dp =
b2 X 1012 exp(- Q/kT)/s, and Q = 0.56 eV, one obtains, from D4.23),
yp/(a° - aD) = 1.09 X Ж1/Mpas, D4.24)
for a = 55 nm at T= 293 °K. The activation energy Q has been determined
from Fig. 44.6. The experimental value of the slope corresponding to this
situation is about 0.66 X 10~7/Mpas. This argument is satisfactory. It has
been similarly found that D4.22) accounts for the observations of stationary
creep in Al-Si crystals (Mori and Osawa 1979). Okabe and Mori A979) have
extended the above analysis to the situation of rod shaped inclusions in
conjunction with the experiment on a Cu-mullite alloy. Again, in this case a
satisfactory agreement is found between the calculation and the observation.
Interfacial diffusional relaxation
Relaxation of the internal stress induced by plastic deformation in a material
with plastically strong inclusions has been discussed on the basis of the
diffusion of atoms through the cores of the dislocations at the matrix-inclusion
interface. This is an adequate picture, since the existence of these dislocations
at the interface is responsible for the internal stress. However, if the stress is
brought about by some other reason, it is better to seek another path of
diffusion such as interfacial diffusion. Interfacial diffusion especially applies
in the situation where an external stress is present in a specimen which has a
so-called incoherent inclusion. The origin of the stress is independent of the
situation at the interface and the atomic arrangement at the matrix-inclusion
interface is in disorder, so the diffusion along the interface is presumably fast.
Even if an interface is described by an array of dislocations, such as a tilt or
twist grain boundary, and atoms are considered to diffuse through pipes of
these dislocations, the diffusional process is adequately treated as interfacial

44. Diffusional relaxation of internal and external stresses
415
(a)
tb)
Fig. 44.7. Change of the misfit due to the interfacial diffusion.
diffusion. In the following, we will discuss the relaxation of an external stress
due to diffusion at the matrix-inclusion interface.
For simplicity, a cylindrical inclusion of radius a lying parallel to the x2
axis is considered. The matrix and the inclusion are elastically isotropic with
the respective shear moduli of ц and /z*. The Poisson ratio of the matrix is v.
Let the applied stress far from the inclusion be o^ = a0. Let atomic diffusion
take place along the matrix-inclusion interface. After diffusion, the resulting
stress field can be calculated by considering a uniform eigenstrain of At^ in
the inclusion. Ас%г is a misfit strain caused by either an mclusion geometry
change or a matrix geometry change due to the interfacial diffusion. These
changes are illustrated by Figs. 44.7(a) and 44.7(b), both of which give the
same misfit Ae^. The dotted line in Fig. 44.7(a) is the shape of the mclusion in
the stress free state, while the solid line in the figure is the void in the matrix
after the mclusion is removed. This is the situation when the diffusing species
are the material constituting the inclusion. Figure 44.7(b) applies to the
situation when the diffusing species are the matrix material.
The eigenstress Aatj caused by Aefj in a cylinder is calculated from B2.11)
as
D4.25)

416 Chap. 7 Material properties and related topics
where
S = S3131 = C-4i0/{8A-i0} D4-26)
and
g = 2(/i*-,i)S + /i. D4.27)
The increase in the elastic strain energy, AW*, is given by B5.31) (a,* = Aa3l):
AW* = - \b,tJ AefjV0 = 2MM*A - 2S){Ai'nfv0/g, D4.28)
where Vo is the volume of the inclusion per unit length, ma2. The applied stress
in the inclusion o* is calculated as
°? = «V/?. D4.29)
Thus, the change in the external potential due to Ae^ is obtained from B5.32)
as
AV=- ofjAifft = - 2a°^A^V0/g. D4.30)
The total energy change or the Gibbs free energy change AW due to Ae^ is the
sum of D4.28) and D4.30),
AW= 2wi*(l - 2S)(Ae^JV0/g- 2aV* A^V0/g. D4.31)
Thus, the misfit strain A7${ which minimizes the Gibbs free energy is found to
be
D4.32)
In this situation, the internal stress Aa31 is given by
A^-^-oY/g D4.33)
from D4.25). Thus, the total stress in the inclusion vanishes,
o? + A^[ = 0 D4.34)
from D4.29) and D4.33). It should be noted that the equilibrium condition,

44. Diffusional relaxation of internal and external stresses 417
D4.34), is a special case of more general conditions. Let the applied stress in
an inclusion be of* and the internal stress be AatJ which is brought about by
the eigenstrain Aefj due to the interfacial diffusion. Then, the Gibbs free
energy is, generally,
AW = - \Aotj AefjVo - a,Aj AefjV0 D4.35)
which is minimized when
( )) = 0, D4.36)
since Асу is linearly related to Aefj. In diffusion, mass is conserved. It can be
expressed by no-dilation,
Atft = 0. D4.37)
From D4.36) and D4.37), it is found that at equilibrium the total stress in the
inclusion becomes hydrostatic,
4a,7+< = S,7a D4.38)
which includes D4.34) as a special case. It is further noted that the above
discussion and conclusion are identical to those developed for a misfitting
incoherent particle in Section 26.
One can discuss the kinetics of the above diffusional relaxation in a simple
manner. Fig. 44.1, the number of atoms which have diffused to accomplish the
formation of misfit Ac^ is calculated as
N = 2V0 Ae^/vQ D4.39)
per unit length of the cylindrical inclusion. Here, Q is the atomic volume. N is
the volume of region I divided by J2, where the major and minor axes of the
ellipse in Fig. 44.1 are a(l + Ae^) and a(l — Ае%г), respectively. When the
relaxation is incomplete and the misfit strain changes by 8(Ае$г), the change
in the Gibbs free energy is given by
S(AW)=-(All1-A^1L^(l-2S)V08(A^1)/g D4.40)
from D4.31) and D4.32). The number of the transported atoms for this
process is given from D4.39) as
D4.41)

418 Chap. 7 Material properties and related topics
Since the diffusion distance / is - \чта, the force F acting on an atom is
calculated as
F = -8{AW)/18N = (АЦг- А<ЪLц*11A - 2S)i2/ag D4.42)
from D4.40) and D4.41). The Einstein relation gives the drift velocity v of an
atom as
v = FDJkT, D4.43)
where Ds is the interfacial diffusion constant, к the Boltzmann constant and T
the temperature. Let the thickness of the interfacial diffusional layer be h.
Then, the number of the atoms which diffuse during the time interval St is
given by
SN = 4hv8t/Q, D4.44)
since the cross section of the diffusion path is 4Л per unit length of the
cylinder. From D4.41) through D4.44), one obtains the rate of the misfit strain
as
d(A^n)/dt = (АЩ - Aeb)/r, D4.45)
where the relaxation time т is given by
D446)
7
Щ
or by
C-4")\^Zl D447)
from D4.26) and D4.27), and VQ = ira2. Since e^ is related to the eigenstrain
of the equivalent inclusion, е|г, in a manner similar to D4.3), and since the
macroscopic deformation is expressed by D4.2) when the volume fraction of
the inclusion / is small, one expects the specimen to respond to the applied
stress by
= (y-y)/r D4.48)

44. Diffusional relaxation of internal and external stresses
419
with т given by D4.46) or D4.47). Here у is the macroscopic shear strain
caused by interfacial diffusion, and
D4.49)
from D4.32). It is clear that, due to diffusional relaxation, the specimen
behaves as a standard linear solid, D4.48) and D4.49), the relaxation strength
of which is by definition (Zener 1948) given by
D4.50)
The preceding treatment of relaxation from interfacial diffusion followed
Mori, Okabe and Mura A980) who also have discussed spherical inclusions.
Koeller and Raj A978) have considered relaxation in a situation exactly the
same as above. However, because they take a different approach, the result is
not exactly the same, although the conclusions are similar in nature.
The interfacial diffusional relaxation has been successfully examined by
internal friction measurements of Al-Si, Cu-SiO2 and Cu-Fe alloys by
Ю"-
5«K>»
10й-
5*0»
0-088«V 10"-
ю"-
г«ю"
Q-I.00W
Average Particle Radius, a (ran)
Fig. 44.8. Particle size dependency of peak temperature Tp.

420
Chap. 7 Material properties and related topics
1.4
Tp (IO"J/K)
Fig 44.8 1 Internal friction of a Cu alloy having Fe particles is measured as a function of
temperature The logarithm of the peak frequency wp times Tpa3 is plotted against the reciprocal
of the internal friction peak temperature Tp, confirming equation D4 47)
Okabe et al. A981), Mori et al. A983) and Monzen et al. A983). The internal
friction was measured as a function of temperature and showed a peak at a
particular temperature Tp, depending on frequencies and particle sizes. The
second phase particles (Si, SiO2 and Fe) in these alloys are considered as
spherical. The relaxation time has the same particle radius dependency as
shown by D4.47). The maximum internal friction occurs when the frequency
up times the relaxation time т is umty. т is also a function of temperature T,
where Ds = DQ exp(-Q/kTp). The relation тир = 1 is plotted by solid lines in
Fig. 44.8 and Fig. 44.8.1 to show the particle size dependency. The slope of the
lines in Fig. 44.8 is -3 since т ос a3. The experimental values plotted in these
figures show a good agreement with the theory.
McClintock A968) has suggested that cavities nucleate on interfaces by
tearing the inclusion away from the ductile matrix or by the cracking of
non-deformable inclusions. The incompatibility (misfit) between the inclusion
and the matrix is expressed by the distribution of dislocations D3.18). These
dislocations are dissipated by punching from the inclusion to initiate the
interfacial cavities or cracks. This idea was first proposed by Ashby A966) and
further developed by Argon et al. A975), Argon A976), Raj and Ashby A971,
1972,1975), Raj A975,1976) and Chuang et al. A979). It is interesting to note
that condition D4.34) is equivalent to the condition of cavity formation.

45. Average elastic moduli of composite materials
45. Average elastic moduli of composite materials
421
The evaluation of average elastic moduli (the shear modulus and bulk mod-
modulus) of composite materials or polycrystals is one of the classical problems in
micromechanics. The pioneer works on this subject have been done by Voigt
A889) and Reuss A929). The Voigt approximation gives upper bounds and
the Reuss approximation gives lower bounds of the average elastic moduli, as
proved by Boas and Schmid A934), Bishop and Hill A951), and Hill A952)
(see also Bruggeman 1934, Huber and Schmid 1934, and Boas 1935).
The Voight approximation
Let us consider a composite material which consists of inhomogeneities with
various shear moduli px, /i2, ••.,/*»! and a matrix with the shear modulus ju0
(see Fig. 45.1). The average (or effective) shear modulus jl of the composite
material will be investigated here. The volume fractions of the inhomogeneities
are denoted by съ с2,...,с„ and that of the remainder (matrix) by c0. The
shapes of the inhomogeneities are arbitrary.
Let us assume that the average elastic strain of the composite material is e°2
when a shear stress is applied. Voigt A889) approximated the average stress of
the composite material as
D5.1)
Fig. 45.1. Composite material containing inhomogeneities 0,, fi2.. ..ju: shear modulus K: bulk
modulus.

422 Chap. 7 Material properties and related topics
which assumes that all the composite elements are subjected to the same
uniform strain ?°2. The definition of the average shear modulus is
Д = о/у, D5.2)
where y = 2€1°2. D5.3)
Then, the Voigt average shear modulus is
D5.4)
The Voigt approximation can also be applied to anisotropic inhomogenei-
ties. Since an = 2C1212«°2 ап& о = fD au dl>, where D is the unit volume, the
average of C1212 becomes
Cl212=
where C^n IS tne elastic modulus in the r-th inhomogeneity with reference to
the same coordinate system in which cf2 IS measured.
The average shear modulus Д of a polycrystal can be evaluated from the
anisotropic elastic moduli Cijkl of constituent single crystals. The single
crystals are assumed to be randomly oriented with respect to the x, y, z axes
shown in Fig. 45.2. The average shear modulus is isotropic and becomes, from
D5.4),
Д = A/8772) f2" d/3 Tsin Odd [2"СШ2 d<>, D5.6)
where CI212 is referred to the x, y, z axes and (/?, в, ф) are the Euler angles.
A random orientation of a single crystal is represented by a point on the unit
sphere shown in Fig. 45.2. The orthogonal crystallographic directions of the
crystal are expressed by the 1, 2 and 3 directions in the figure. The location of
the point on the unit sphere indicates the 3-direction, and angle ф from the
median indicates the 1-direction, where the plane containing the 1- and
2-directions is tangential to the unit sphere. The surface element of the unit
sphere is sin в d/3 &0 and the total area is 4ir. Angle ф varies from 0 to 2w.
Therefore, A/8тг2) sin в d0 d0 йф in D5.6) corresponds to cr in D5.5). The

45. Average elastic moduli of composite materials
423
Fig. 45 2. A random orientation of a single crystal is represented by a point of the unit sphere The
crystalline directions are expressed by the axes 1, 2, 3
elastic modulus C1212 in the x, y, z coordinate system and CIJKL in the 1, 2, 3
coordinate system are related by
1212 ~ a\lalJa\Kall>'IJKL->
D5.7)
where
i cos в cos /J cos ф — sin (8 sin </>, - cos 0 cos /? sin ф - sin /} cos ф, sin б cos /J
cos в sin 0 cos ф +cos /8 sin ф, -cos 0 sin /} sin ф +cos /} cos ф, sin в sin /8
^ — sin 0 cos ф, sin в sin ф, cos б
D5.8)
The average bulk modulus К can be obtained in a similar way. The bulk
moduli of the inhomogeneities are denoted by Къ К2,---,Кп and that of the
matrix by Ko. The Voigt approximation is obtained by assuming that all the
elements of the composite material have the same dilatation c° as the average.
Then, the average hydrostatic stress a is evaluated by
a=
D5.9)
where
D5.10)

424 Chap. 7 Material properties and related topics
By definition, the average bulk modulus is
К=а/в. D5.11)
Therefore, we have
n
K= I crKr. D5.12)
r = 0
The average bulk modulus of a polycrystal is evaluated in the same way.
Since au = \CHjJ6, and a = \$D a,, dD, we have
1ШвйВ, D5.13)
where D is the unit volume. The quantity CHjJ is an invariant and is
independent of /?, в, ф in D5.6). Therefore, D5.13) is simply а=\Си^в,
which leads to
K=\CtiJJ. D5.14)
The Reuss approximation
Reuss A929) has assumed that all the elements of the composite material (Fig.
45.1) are subjected to a uniform stress equal to the average stress. According
to Reuss' assumption, the average shear strain for a given shear stress a = a°2
is expressed by
У= Icra/Mr. D5.15)
r = 0
By definition D5.2), the average shear modulus is obtained as
. D5.16)
Similarly, the average bulk modulus becomes
. D5.17)
=o

45. Average elastic moduli of composite materials
425
Table 45 1 (after Hill 1952)
Al
Cu
Au
a-Fe
1.08
170
186
2.37
Си
0 622
1.23
1.57
141
С*
0 284
0 75
0.42
116
GR
0.26
0.40
0 24
0 74
Gv
0 26
0 54
0 31
0.89
KR — Kv
0 78
1.39
1.67
173
Er
0 71
1.09
0 69
1.93
Ev
0.71
144
0 87
2.29
"R
0 349
0 369
0.431
0.313
"v
0 348
0 328
0 413
0.280
Unit=1012dyn/cm2.
For a polycrystal, we have
Д = \(l/2n2) Г' dp Tsin в d0/%212
I •'о Л) •'о
D5.18)
where sijkl is the elastic compliance and у = 2e12 = ^s^^u- The average bulk
modulus obtained by the Reuss method is identical to the Voigt result for the
following reason: since в = «;,• = sHjJa for the hydrostatic stress a and sUjj is
an invariant, it follows that в = sHjjd and K= \/snjj = CUjJ/9.
Table 45.1 (Hill 1952) contains numerical results of ju and К for various
polycrystal metals. The Voigt and Reuss approximations for /I are denoted by
Gy and GR, respectively. The Voigt and Reuss approximations for К are
identical and are denoted by Kv or KR. The average values of Young's
modulus and Poisson's ratio are evaluated from ju and К by using the well
known relations (A2.2). Ctj in the table are the elastic constants of the single
crystals. Table 45.2 shows the experimental observation. According to Hill
\{GR + Gv) or (GRGVI/2 are good approximations since
GR < G < Gy,
D5.19)
where the true value of ju, is denoted by G.
Table 45 2 (after Hill 1952)
Al
Cu
Au
a-Fe
G
0 265
0.436
0.278
0.808
К
0 74
133
166
159
E
0.71
1.18
0 79
2 07
V
0.34
0.35
0.42
0 285
Unit = 1012 dyn/cm2

426 Chap. 7 Material properties and related topics
Hill's theory
Hill A952) has shown that the Voigt approximation and the Reuss approxima-
approximation are the upper and lower bounds of the true average elastic moduli.
Let the average stress and strain in an aggregate of single crystals under an
applied force X-, be denoted by a,y and ltj. a0 and ltj are uniform quantities
which can be calculated from the boundary values. Namely, otJ is estimated
from
Oijn^Xi on S, D5.19.1)
where иу is the unit exterior normal of the material surface S. lu is obtained
from the equation
сi, — \{п; , + m, ,) in D,
" K ''J J''' D5.19.2)
u, = u° onS,
where м, is the average displacement which is a linear function of the space
coordinates. _
Denoting the average elastic moduli by Ci]kh we can write
% = Cijklik, D5.19.3)
or
iij = Sijki°kh D5.19.4)
where sijkl is the average elastic compliance tensor [see (A2.18)']. The actual
stress, strain, and displacement are denoted by ajy, «,y and uh respectively.
They must satisfy the same boundary conditions as in D5.19.1) and D5.19.2),
а, и = Xj
„ onS. D5.19.5)
u, = w,u
Since Ojj j = 0, cijj j = 0, we have
o,j4] dD = f aljUiJ AD = fxiU° dS,
Jd Js D5.19.6)
jitj dD = j^ju^ dD = fxtf dS.

45. Average elastic moduli of composite materials 427
Therefore, it holds that
aIJiiJ = (l/V) J4ye,7 dD, D5.19.7)
where V is the volume of D. New stress a,° and strain e°7 are defined by the
following: ofj is the stress that would exist in a crystal with a local orientation
and having the strain itJ. e,7 is the strain that would be produced in such a
crystal by stress a,7. Namely,
f=^" D5.19.8)
?ij ~ Sijklakh
where Cijkl and sijkl represent the elastic modulus and the compliance of the
single crystal. We have
°u = Cijki*ki> <¦>, = sijki°ki- D5.19.9)
From D5.19.8) and D5.19.9), a°J€u = aiJiij and а,7«°• = аое,7, and therefore,
°u€U + (a<7 - ff°)(?'V - ?"o) = a/°«O + 2(?'7 ~ ё/у)а0, ^ ^ ^
+ ( ){ Ь) 5Л + 2( d)
The second terms in the left-hand sides in D5.19.10) are positive since they
can be written as Cljkl(ekl- ?*,)(e,., - iu) and sijkl(okl - дк,)(аи - а,7), re-
respectively. It is also shown that
/ a,7(e,7 - itJ) dD = ГхДм,0 - и?) d5 = 0,
Д ^ D5.19.11)
а,7 - a,>,7 dD = Js(Xt - Xt)u* dS = 0,
by integrating by parts. Then, D5.19.10) leads to
Vaij€iJ<ilJfa°dD,
D5.19.12)

428 Chap. 7 Material properties and related topics
where D5.19.7) has been used for the derivation. Thus, we have, finally,
Cljkl<{\/V)jCljklUD
D5.19.13)
This result indicates that the Voigt approximations give upper bounds and the
Reuss approximations yield lower bounds, since 4j1212 = 1/C1212, and sUJ] =
9/С1Ш.
Eshelby's method
Let us consider again the composite material shown in Fig. 45.1. We evaluate
the average shear strain у when an average shear stress is given as o®2 = o.
When the relation between у and a is obtained, the average shear modulus is
determined by D5.2).
The stress in the composite material varies from place to place. Here it is
denoted by a°2 + o12, where au is the disturbance due to inhomogeneities.
Eshelby's equivalent inclusion method in Section 22 is employed. The com-
composite material is simulated by a homogeneous material having the same
elastic moduli as those of the matrix and containing inclusions &2r with
eigenstrain ef2, where r=l, 2,...,«. ur is the r-th inclusion with the same
volume and location as the r-th inhomogeneity. The equivalency equation
B2.5.1) is
°12 + *12 2Mo(«12 + «12«2) .
inJ2r D5.20)
= 2мДе1и2 + ?12),
where
€12 = 251212€г*2, а1°2 = 2/х0?1°2, D5.21)
and Sijk, is Eshelby's tensor; S1212 = D - 5po)/15A — v0) for spherical inclu-
inclusions. The equivalent eigenstrain e^ in Qr is obtained from D5.20) as
D5-22)
where
D5.23)

45. Average elastic moduli of composite materials 429
The following quantities in ?2r are easily calculated by substituting D5.22)
into D5.21) and D5.20):
e12 = 2Suu(ti0 - nr)e°2/gr,
an = 2BSUU -
о
-Е
is added to e?2 + e12 in D5.24). The average strain is
D5.24)
According to Tanaka-Mori's theorem D2.20), we have
f ol2dD = 0 D5.25)
JD-a,
and, therefore,
n
fa12dD= LcrB51212-l)(Mo-Mr)a1o2/gr. D5.26)
This does not, however, satisfy the condition for stress disturbance
(ol2dD = 0. D5.27)
JD
Therefore, we add the uniform stress
- E c,BSnu - l)(/io - Pr)°i2/gr, D5.28)
r=\
to an in D5.24). Consequently,the uniform strain
crBS12U - l)(ju0 - /1г)аг°2/2мо«, D5.29)

430 Chap. 7 Material properties and related topics
n n
= ? cra/gr - Y, crBsi2u - 1)(Мо - /0°//*о?,- + coYo> D5.30)
r=l r=l
where y0 is the average strain in the matrix, and
Yo = a/^o. D5.31)
By using definition D5.2) and
n
co+ Ecr=l, D5.32)
we have
1+ ЕсДмо-Мг)/{Мо + 251212(мг-/хо)} . D5.33)
Similarly, the average bulk modulus is obtained as
K= Ko/\ 1 + I cr(K0 - Kr)/{ Ko + \SUjj(Kr ~K0)}\. D5.34)
The Reuss approximation is a special case, 2Sm2 = 1. The Voight approxima-
approximation can also be obtained by 2S12U = 0 as well as by expanding the denomina-
denominator, and assuming that the second term is much less than 1.
The results D5.33) and D5.34) can be obtained from the energy considera-
consideration of Section 25. The elastic strain energy of the composite material is
written as B5.11). On the other hand, we can write it as
W* = \d2/]L. D5.34.1)
Substituting D5.22) and D5.34.1) into B5.11), we have
П
,- D5.34.2)
r=l
Then, it leads to D5.33).
Self-consistent method
When the equivalent inclusion method is applied to obtain the stress dis-
disturbance of inhomogeneities, we assume the existence of a matrix which has

45 Average elastic moduli of composite materials 431
the shear and bulk moduli ju0 and Ko. This assumption, however, does not
hold in some cases. Polycrystals, for instance, have no matrix, although they
are aggregates of particles with different elastic moduli. In composite materials
as shown in Fig. 45.1, the meaning of the matrix becomes rather vague when
the volume fraction of inhomogeneities is increased, particularly when c0 -> 0.
Besides this, the interaction among inhomogeneities becomes more prominent
as the number of particles increases. Kroner A958) and Budiansky and Wu
A962) have proposed the self-consistent method in order to avoid the difficul-
difficulties mentioned above.
Let us consider a composite material as shown in Fig. 45.1. The average
strain and stress under an applied shear stress field are denoted by e°2 and a°2,
respectively. Hooke's law is
o?2 = 2]ie°2, D5.35)
where Д is the average shear modulus. The stress disturbance due to the
inhomogeneities is denoted by a12. The self-consistent method assumes that
the composite material can be simulated by a homogeneous material which has
a shear modulus Д, Poisson's ratio v, and an eigenstrain e*2 in ?2r (r =
1,2,..., и). The equivalency equation B2.5.1) becomes
2 + €12?12) .
in Or D5.36)
= 2/xr(e1u2 + e12),
where
€i2 = 2S1212?f2. D5.37)
The Poisson's ratio appearing in S1212 is the average Poisson's ratio v. e%2
satisfying D5.36) is obtained as
«?2 = (M-/O«i°2/fr, D5-38)
where
?г = Д + 2?1212(Мг-Д). D5.39)
The strain and stress in ?2r are obtained as
,, D540)
u/

432
The
r =
average
o
•'D
values
+ e12).
are
n
r=l
n
Chap. 7 Material properties and related topics
D5.41)
_ Г / Q \ ~-,
where a0 and y0 are the average shear stress and strain in the original matrix
with shear modulus ц0:
ao = /W D5.42)
Eliminating a0 from D5.41) and D5.42), we have
n
o = Mo?+ L cr(nr-ix0)jiy/gr, D5.43)
and, therefore,
n
ji = fto+ ? cr(/ir —/to)ju/{/Z + 25'1212(/*г — ju)}. D5.44)
Similarly, the average bulk modulus is obtained as
n
K=K0+ ? cr{Kr- K0)K/[K+$StiJJ(Kr- K)}. D5.45)
In the case of spherical inclusions, we have
The relations in (A2.2) give
v=CK-2Jl)/2(ji+3K). D5.47)
Equations D5.44) and D5.45) with D5.46) and D5.47) are simultaneous
equations to determine /I and К for given cr, цг, Kr (r = 0, 1,..., и). These

45. Average elastic moduli of composite materials 433
equations are valid even if c0 = 0. In this case, ju0 and Ko are taken as zero in
D5.44) and D5.45). Equations in D5.41) for c0 = 0 lead to
n n
1 = I criir/gr, 1 = ? crJL/gr- D5-48)
The above relations were pointed out by Budiansky A965).
T.T. Wu A966) has solved simultaneously D5.44) and D5.45) and has
investigated the shape effect of inclusions when Eshelby's tensor Sijk, for
spheroids is used. He concludes that the disk-shaped inclusions give by far the
most significant increase in Young's modulus. Walpole A967) has extended
this theory to include the anisotropic case and has proposed a convergent
series of successive approximations. Kneer A965) and Morris A970) have
calculated elastic moduli of polycrystals where the orientation distribution of
single crystals is expanded in a series of generalised spherical harmonics.
Hershey's paper A954) may be one of the oldest ones, after Voigt and Reuss,
dealing with the elasticity of a polycrystalline in terms of the elasticity of the
individual grains.
But the equivalent inclusion method, including the self-consistent method,
is not accurate enough when one wants to know, for instance, the effect of the
distribution pattern (morphology) of unidirectional fibers in a reinforced
composite. In order to calculate the interaction among composite elements,
boundary value problems of elasticity must first be solved. Exact analyses
along this line have been attempted by Adams and Doner A967), C.H. Chen
A970), Chen and Cheng A967), and Sendeckyj A970, 1971), among others.
Upper and lower bounds
Hashin and Shtrikman A962, 1963) have proposed a vanational principle for
finding upper and lower bounds on the average elastic moduli of a composite
material. Their method gives generally better bounds than the Voigt and Reuss
bounds. Their method has been generalized and modified by Hill A963),
Walpole A966, 1969, 1970), Willis A977), Willis and Acton A976), Kroner
A977), and Laws and McLaughlin A979), among others.
Let us consider an inhomogeneous system with various elastic moduli Ci]kl.
The overall (average) elastic moduli are denoted by Cijkl. The strain and stress
state under an applied load is simulated by the equivalent inclusion method.
The inhomogeneous system is simulated by a homogeneous system with
eigenstrain t* and elastic moduli Cfjkl. The average strain is denoted by e°7.
The equivalency equation B2.5.1) becomes
Q%/ («°/ + 4, ~ «2/) = ClJkl (& + €*/). D5.49)

434 Chap. 7 Material properties and related topics
where ekl is the strain disturbance due to the inhomogeneities, expressed by
C.24) in terms of C°/m/Ie* „. Let us define a polarization stress a* by
°?j = C°klt*kl. D5.50)
Expression C.24) can be written symbolically as
е = Го*. D5.51)
Then, D5.49) is written as
а* + (С-С°)(?° + Га*)=0 D5.52)
or
(С-С°)о* + (?° + Га*) = 0. D5.52.1)
By the use of definition
(/. g) = (VV)fj(x)g(x) Ax, D5.53)
Hashin and Shrikman A962) have proposed the variational principle
=0, D5.54)
where the comparison function is a*. The stationary equation for the variation
becomes D5.52.1) and its stationary value is
К"*, <°) = \f C,V^°, AD = \(о0к1е*к1 AD, D5.55)
JD JD
where a° is defined by
o° = C°k,4,. D5.56)
The stationary value of the functional can be written, from B5.11), as
W*-W°, D5.57)
where W* is the elastic strain energy and W° is \jD o°e°j AD.

45. Average elastic moduli of composite materials 435
It can also be shown that the stationary value is a minimum when С — C°
is positive definite, and a maximum when С — C° is negative. That is,
< |(а*, (С- С0)"^*) + $(„*, Га*) + (а*, е°) D5.58)
for a positive definite С - C°, and
> |(а*, (С- С0)-^*) + |(а*, Га*) + (а*, с0) D5.59)
for a negative definite C— C°. When а* is approximated as a linear func-
functional of e° in the variational method D5.54), the inequalities D5.58) and
D5.59) can be used for estimating the lower and upper bounds of С by
choosing two sets of C° so that C— C° becomes positive or negative definite,
respectively. The minimum or maximum of the functional in D5.54) depends
on the operator (С- С0) + Г. Willis A977) has shown that the operator is
positive definite so long as С — C° is positive definite at each point of D, and
is negative definite so long as С — C° is negative definite at each point of D,
as follows. Suppose that С — C° is positive definite. Then,
(*, (C-C°)x)>0 D5.60)
for any non-zero x. Let (C-C°)x = y. Then, x = (C-C°)~1y; therefore,
(x, (C-C°)x) = (x, y) = (y, x)= (y, (C-C°y1y)>0. D5.61)
This means that (C - C°)~l is positive definite if C- C° is positive definite.
We know that for any e* on a*, equation
a = C°(e-e*) = C°€~a* D5.62)
satisfies the condition aijj = 0 and (а, с) = 0 as long as e is obtained from
D5.51). We have, therefore,
(а*, Га*) = (а*, е) = ((C°e - a), e) = (C°e, e) > 0. D5.63)
This means, then, that Г is positive definite.

436 Chap. 7 Material properties and related topics
If С — C° is negative definite, we may consider the following approach.
From D5.50) and D5.62),
e* = (C°)~V e = (C°)~1a + €*. D5.64)
Hill A963) calls e* the strain polarization. It now follows that
(а*, Га*) = (C°e*. в) = (CV, {(C°yla + e*))
= (e*,CV)-(a,(C0)"la), D5.65)
since (e, a) = 0. Consider the following identities:
c°{c°yl = cc-\
D5.66)
cQc-l = c°c-1.
The difference of the above two identities leads to
(C- C°)~lC°= (CT'^C0) - C) - /, D5.67)
where / is the identity. Thus,
(a*, (C- C0)^*) = (a*, (C0)^0) - С-^Л*) - (а*, €*),
D5.68)
and, therefore,
(a*, (C- C°)~ V) = (e*, ((C0) - C-1)"^*) - (e*, CV). D5.69)
The sum of D5.65) and D5.69) provides
(о*,(С-С°У1о*) + (о*,То*)
= -(**, (С - (С0))""V) - (a, (C0)^). D5.70)

45. Average elastic moduli of composite materials
437
© ,
- //
//
I!
Reuss
Hoshin-Shtrikman't|
upper bound
Haihin -Shtrihman'*]
lower bound
Eshelby
Self-consistent
method
04
0 5 c,
Fig. 45 3. The effective shear modulus versus the volume fraction of spherical inhomogeneities,
=100.
If (C-C°) is negative definite, then (C - (C0)) is positive definite.
Therefore, the above equation D5.70) proves that the functional of a* in
D5.59) is negative definite.
Some numerical examples are shown in Figs. 45.3 and 45.4 for a composite
material containing spherical inhomogeneities. The ratios between the elastic
isotropic moduli of the spheres and the matrix are taken as juj/jUq = 100,
Кг/К0 = 100. Poisson's ratio is taken as v1 = v0 = 0.3. The average shear
modulus and bulk modulus are shown as functions of the volume fraction cx
of spheres. For comparison, the Voigt bounds D5.4) and D5.12), the Reuss
bounds D5.16) and D5.17), Eshelby's approximations D5.33) and D5.34), and
the self-consistent approximations D5.44) and D5.45) are also shown in the
figures.
Other related works
Many papers have been published for predicting elastic properties of fibrous
composites. The review papers by Hashin A964) and by Chamis and Sendec-
kyj A968), for instance, provide comprehensive lists of the literature. Various

438
Chap. 7 Material properties and related topics
к/к0
i /
Ф Voigt
@ Reuss
C) Hoshin-Shtrikmon's
upper bound
(§) Hathin - Shtrikman's
lower bound
© Eshelby
© Self-consistent
method
®
0.4
0.5 C,
Fig. 45.4 The effective bulk modulus versus the volume fraction of spherical inhomogeneities,
k,/a:0 = ioo
theories have been proposed which can be classified, according to Chamis and
Sendeckyj, as netting analysis, mechanics of materials, self-consistent model,
variational, exact, statistical, discrete element, semi-empirical models, and
theories accounting for microstructure.
For many steady-state and transient dynamic problems of structural mech-
mechanics, it is helpful and sometimes required to know the dispersion characteris-
characteristics of free harmonic waves, i.e. the dependence of the phase velocity as the
wavelength. For waves propagating in the direction of the fibers, a fiber-rein-
fiber-reinforced composite is essentially a system of wave guides. Achenbach and
Herrmann A968) have derived the dispersion curves for transverse time-
harmonic waves. Further dispersion curves are obtained by means of a
smoothing operation (Sun, Achenbach and Herrmann 1968), and by varia-
variational methods (Kohn, Krumhansl and Lee 1972, Wheeler and Mura 1973,
Nemat-Nasser and Minagawa 1975, and Nemat-Nasser and Horgan 1980).
The self-consistent imbedding approximation has been applied to rocks and
porous media by many authors: Walsh A968, 1969), Korringa A973), Ander-
Anderson et al. A974), O'Connell and Budiansky A974), Budiansky and O'Connell

46. Plastic behavior of polycrystalline metals and composites 439
A976), Edil et al. A975), and Korringa and Thompson A977). When some
cracks close and when some closed cracks undergo frictional sliding, then the
overall response becomes anisotropic and dependent on the loading condi-
conditions, as well as on the loading path. The self-consistent method is used to
estimate the overall moduli by Horii and Nemat-Nasser A983).
Another important field of micromechanics is the mechanics of granular
materials (soil, powder, rockfill). The works of Dracker and Prager A952),
Shield A953, 1954), Mindlin A954), and Mandl and Luque A970), are
phenomenological continuum theories. The microstructural continuum theo-
theories for granular materials have been developed by Mandel A947), Spencer
A964), Home A965), Mogami A965), Oda A972), and Mehrabadi and Cowin
A978), Mroz A980), among others (see Cowin and Satake 1978). An interest-
interesting application of the Cosserat continuum A909) to granular materials has
been given by Oshima A953), Mindlin A964), Eringen A966), and Satake
A978). For finite plastic deformations of porous metals and geotechnical
materials such as cohesionsless or cohesive soils, Nemat-Nasser and Shokooh
A980) have developed a theory which accounts for plastic volumetric changes,
pressure sensitivity, microscopic frictional effects, and a nonassociative flow
rule.
46. Plastic behavior of polycrystalline metals and composites
Taylor A938) has evaluated the flow stress of a polycrystal in tension from the
critical shear stress required to cause slip in a single crystal. He regards the
polycrystal as an aggregate of randomly oriented FCC crystals that are
assumed to be rigid-plastic. Hershey A954) and Т.Н. Lin A957) have consid-
considered the influence of elastic strains on the Taylor theory. They have assumed
that the total strain is the same in each grain (single crystal) and outlined a
procedure which would predict slipping in different slip systems within a grain
with an increase in the prescribed total strain. Later, Kroner A961), Budiansky
and Wu A962), Lin A964), Hill A965), and Hutchinson A970), among others,
have abandoned the assumption of a homogeneous strain throughout the
crystal and used instead the self-consistent method. These and related topics
are reviewed in this chapter.
Taylor's analysis
The first realistic model for calculating the polycrystal uniaxial stress-strain
relation from that of the single crystal was proposed by Taylor A938). An
aggregate of randomly oriented crystals under tension is assumed to be

440
Chap. 7 Material properties and related topics
Fig. 46.1. FCC single crystal.
rigid-plastic. The associated average stress is denoted by atJ and the average
plastic strain by Ifj.
Suppose that grains of the polycrystal are FCC single crystals which have
various orientations. Consider an arbitrary single grain. The relative orienta-
orientations of the four slip planes of the FCC crystal are those found in the faces of
a regular tetrahedron, and the three possible slip directions in each plane
coincide with the edges of the triangular faces. The orientation of the tetra-
tetrahedron abed, relative to the cubic crystal axes x, y, z is shown in Fig. 46.1.
Relative to an arbitrary set of Cartesian axes 1, 2, 3, the «th slip system
(n = 1, 2,..., 12) may be specified by the unit vector /nj"' normal to the slip
plane, and the unit vector n\n) in the slip direction.
Consider a uniform stress state atj in a crystal of a given orientation. The
resolved shear stress on the nth slip system is
and hence its rate is
*<">=6U^\
where
D6.1)
D6.1.1)
D6.2)

46. Plastic behavior of poly crystalline metals and composites 441
The plastic strain in the grain due to the slips that have occurred along all the
slip system is
'fj= Еу("Чв). D6-3)
and its time derivative
«X-= ?v(n4n)> C46-4)
where у(п) is the amount of plastic shear strain associated with т(п), the
summation being taken over all active slip systems.
Taylor assumes that the plastic strain and its time rate are homogeneous
and independent of the orientations of grains; in other words,
tfj = i?j, ifj = cfj- D6.5)
The plastic work done on all the crystals is
D6.6)
'и = 1
where D is taken as a unit volume. Taylor further assumes that the strain
hardening law chosen to represent the single crystal behavior, is
D6.7)
that is, the flow stress on each system in a grain is equal and is a function of
?n|y(n)|. Taylor's analysis follows the following steps. For a given ifj, y(n)
are calculated from
«?=Ey(b4/b) D6-8)
n
and
? | y(n) | = minimum, D6.9)
n
where a["' varies from grain to grain. The orientations of the grains are
equally distributed in a stereographic projection. After у(л) are found, they are
substituted into D6.7) to evaluate т(п). The average stress atJ is calculated

442
Chap. 7 Material properties and related topics
Strain с, (polycrystal)
1.2 1.3
O.I 0.2
0.3 04 05 06 07 08
Strain, s (single crystal)
0 9 1.0
Fig. 46 2 Theoretical prediction of polycrystal stress-strain curve from a single crystal stress-strain
curve (after G.I. Taylor 1938)
from D6.6) by using ifj, y(n) and т(п). The integral on the right-hand side in
D6.6) is replaced by the integral shown in D5.6), since the orientations of the
grains are equally distributed on the unit sphere associated with the Euler
angles. A numerical example calculated by Taylor is shown in Fig. 46.2, where
the load-extension curve (P ~ e) for a polycrystal aluminum is calculated from
the shear stress-strain (S ~ s) curve of a single crystal aluminum. Points
indicated by О are data for the single crystal, X and + show the experimen-
experimental values of the polycrystal, and points О are the calculated values.
The minimum principle D6.9) has been proved by Bishop and Hill A951) in
the following manner. Let d?,y be a prescribed increment of strain in a single
crystal and let o,y be a stress which will produce this strain by activating a set
of shears dy'"' with critical shear stress тк. Let dy*(n) be any set of shears
which are geometrically equivalent to the prescribed strain, but which are not
necessarily operable by any stress satisfying the yield conditions; that is,
D6.10)
and
D6.11)

46. Plastic behavior of polycrystalline metals and composites 443
Multiplying D6.10) by atj, we have, from D6.1),
?T(») dY<"> = ?т(и) dY*(n). D6.12)
n n
Equations D6.11) and D6.12) lead to
")l D6.13)
which is equivalent to the statement D6.9).
Taylor's hardening law D6.7) has been further studied and modified by
Hutchinson A964), Hill and Rice A972). Havner and Shalaby A977, 1978),
Shalaby and Havner A978), and Havner et al. A979).
Applying the self-consistent method to finite elasto-plastic deformations,
Iwakuma and Nemat-Nasser A984) have estimated the overall moduli of
polycrystalline solids. The method predicts a Bauschinger effect, hardening,
and formation of vertex or corner on the yield surface. Their interesting
conclusion is that small second-order quantities, such as rotations of grains
and residual stresses have a first-order effect on the overall response, as they
lead to a loss of the overall stability by localized deformations.
Self-consistent method
Taylor's analysis has been modified by using the idea of the self-consistent
method explained in Section 45. The stress otJ and strain €tJ in a grain are
different from the average stress ai} and the average strain i^. The difference
of the plastic strain efj and its average value e? is considered the cause of the
non-uniformity. The misfit e? — ifj is considered as the eigenstrain used for
the spherical inclusion problem. The strain caused by the misfit can be written
as Sijki(€ki ~ **/)> where Sijki is tne Eshelby tensor. If the material has no
grains and is homogeneous everywhere, the strain is etJ¦= C~jllakl + ifj. How-
However, when the material has the misfit, we have
«,, = Crjl,okl + Ifj + SiJkl{tl, - lpkl). D6.14)
The first term on the right-hand side of D6.14) is the average elastic strain.
The last term is composed of the elastic and plastic strain disturbances, the
sum of which is the total strain disturbance due to the misfit between e? and
ifr On the other hand, it must hold that
еи=Спи°ы + €и> D6-15)

444 Chap. 7 Material properties and related topics
where atJ is the stress in a grain. The overall average of atj is a,y. The overall
average of e,y is iijy and that of e? is e?. Multiplying the elastic moduli to the
equation resulting from D6.14) = D6.15), we have
a,j = аи + Cijkl { Sklmn{iL-iD-Ы,- Ц,)} D6.16)
or its time derivative
au = 5U + Cljkl {Sklmn{ipmn - lpmn) - {ipkl- ipkl)}. D6.16.1)
The twelve slip systems are considered in a FCC single crystal of a given
orientation. The resolved shear stress on the na slip system can be expressed
by D6.1). The plastic strain in the grain can also be given by D6.3).
т<п) and y(n) are related by the plastic shear stress-shear strain relation of
the single crystal,
т = т(у) огт = т'у. D6.17)
т(л) is the value of т, and 7(n) is the value of у for the nth slip system.
Equations D6.16.1), D6.1.1), D6.3), D6.17), and the following concept of
average are fundamental ingredients in finding oi} as a function of ifr The
average of efj for all possible orientations of single crystals (grains) is Ifj.
Namely,
<O = (^average- D6-18)
Oij starts from zero. atj = atj until т(л) (resolved shear stress) in some grains
reaches the elastic limit ry. If a,y is increased by atj, then e? is increased by ifj
from zero value. In order to find ifj we must solve D6.16.1) by assuming Ifj
from zero value. In order to find if} we must solve D6.16.1) by assuming
ej,, = 0 (initial value). Since rf,7 is also a function of ifj, it is convenient to
rewrite D6.16.1) in another form. Multiplying D6.16.1) by a\"^ (since we know
which grains and which slip systems are plastically active at this moment, we
know a^) we have, from D6.1.1),
f <"> = Soaj;> + *^СШ { Sklmn {ipmn - ipmn) - (e?, - ipkl)}. D6.19)
Furthermore, from D6.4) and D6.17) we have
L, 4 amn €mn i-i У akl ekl
D6.20)

46. Plastic behavior of polycrystalline metals and composites 445
where n = 1, 2,..., 12 when all slip systems are plastically active. Some of the
n slip systems are not active in the initial state after the elastic limit. The
unknown quantities are у(п) in D6.20) (ifj = 0). The solutions of y(n) are
substituted into D6.4) to obtain ifr ifj is not zero for the calculation of the
second stage when a,y is further increased, ifj is calculated from ifj by
tft = A/8772) f2' djS Tsin OdO [2'if.-d4>, D6.21)
Jo •'о Л)
where the Euler angles (/}, в, ф) (Fig. 45.2) relate the orientation of stress axes
A, 2, 3) to that of the crystal axes (x, y, z). The new increment ifj provides
an increment of у(п> in D6.20) by giving new atj and ifj, where atJ is
arbitrarily chosen and ifj is calculated by D6.21) in the preceding calculation.
The same step-by-step calculation is continued. itj is easily calculated by
u fj. D6.22)
A more general plastic shear stress-strain relation is
#<»>= ? nt"m)y(m) D6.23)
m = l
which may be used for the left-hand side of D6.20). In this calculation, we
must keep in mind the loading and unloading conditions as well as the
condition that f(л) and y(n) must have the same sign.
The method of solving D6.20) is essentially the one proposed by Kroner
A961), Budiansky and T.T. Wu A962), Т.Н. Lin A964) and Hutchinson
A970). The method proposed by Hill A965) is slightly different. Equation
D6.15) is written as
aij = Cijkl\ekl ~ €kl)
or D6.24)
°ij= CijkMki-i-ki)-
When D6.24) is multiplied by a\"^, we have
*w = a\?CiJk,{ik,-il,). D6.25)

446 Chap. 7 Material properties and related topics
Substituting D6.23) and D6.4) into D6.25), we obtain
? *(->?<»> + a™CtJkl ? y(m47) = «\?Cimikl. D6.26)
m = l m = \
By solving D6.26) to express у(л) in terms of ikl, we have
Y^W^V D6.27)
Then, D6.24) becomes
О D6-28)
If D6.28) is substituted into D6.16.1), we have
CtJJik,- ? fpl?«№P,) =4 + Cijkl{Sklmn{iPmn-ipmn)
V и = 1 7
D6.29)
From D6.29) we can calculate e? in terms of given values of &и and the initial
value of Ifj in the preceding calculation. The new ifj is calculated from ifj by
D6.21).
The early work by Lin A957) used D6.20) without the second term, in
which he assumed ifj = ifj. Figure 46.3 shows the numerical example obtained
by Hutchinson A970), showing the comparison of various theoretical tensile
stress-plastic strain curves for FCC polycrystals composed of isotropic and
non-hardening single crystal grains. The Lin model refers to Lin's early work
A957). The К. В. W. model stands for the works by Kroner, Budiansky and
Wu. These curves are obtained by assuming that the single crystals have no
work-hardening. In this case the left-hand side in D6.20) is zero after yielding.
The yield shear stress is denoted by т .
Equation D6.20) can also be used for determining the yield surface of a
polycrystal. Lin and Ito A966) and Lin A971) have assumed that a polycrystal
consists of identical basic cubic blocks of 64 cube-shaped crystals having
different orientations. The calculation is similar to the cubic inclusion prob-
problem. Further interesting studies on the yield surface have been done by
Mandel A966), Bui A970), Zarka A972, 1973), and Weng and Phillips A977),
among others.

46. Plastic behavior of polycrystalline metals and composites
447
Fig. 46 3. Vanous theoretical predictions of stress-strain curves of a FCC polycrystal composed of
isotropic and non-hardening single crystal grains (after Hutchinson 1970).
For the first original work to determine the yield function from multi-slip
of an isotropic aggregate of FCC crystals the reader is referred to the two
papers by Bishop and Hill A951). Recently, Bassani A977) has examined the
yield function of FCC (and BCC) metals whose texture gives rise to trans-
transversely isotropic plastic properties.
Chen and Argon A979) employed the self-consistent theory to consider the
effect of grain boundary sliding in creeping polycrystals and composites where
the sliding boundaries are treated as disk-shaped viscous inhomogeneities and
the matrix (grain interior) follows a power law in creep. Raj and Ashby A971)
and Raj A975) considered a similar problem and investigated the nucleation
of intergranular cracks during steady state and cyclic creep. Their analysis is
based upon stress concentrations at inclusions which are produced by grain
boundary sliding.
Ghahremani A980) has investigated the effect of grain boundary sliding on
anelasticity of polycrystalline materials. Recently, Weng A981) has employed
the self-consistent method for predicting the creep and recovery strain of an
aluminum alloy. Weng A984) has given the stress and strain state of con-
constituent phases for a general multiphase, anisotropic composite with arbitrarily
oriented anisotropic inclusions. The self-consistent method has been applied to
composite materials by various investigators (e.g. Taya and Mura 1981, Taya

448 Chap. 7 Material properties and related topics
and Chou 1981, 1982, Takao et al. 1982, other important references in
Christensen 1979, T.W. Chou and M. Taya 1980, and Hashin 1983).
Embedded weakened zone
If a material contains a weakened zone, п, the zone can be simulated by an
ellipsoidal inhomogeneous inclusion. For simplicity it is assumed that the
principal axes of the ellipsoid are parallel to the coordinate axes. The two
phases Q and the matrix have different shear moduli /x* and jn, respectively,
and different resistances for plastic deformation. It is assumed that an applied
stress o®2 causes uniform plastic strain e m ®> an<^ uniform plastic strain ef2
in the matrix. The stress disturbance a,7 can be obtained from B2.19) where
tfj is replaced by (e - tf2). For a single weakened zone, the stress in ?2 is
obtained as
The strain disturbance e/7- in ?2 is
oc (иУ/0(сГ2 4i) + (i /i//*K2/2/i
«12 = 2S1212 , гт-^ • D6.31)
l+2(/t*/M-l)Si212
The total stress and strain in Я are
ar=f;ai2' D6.32)
?12 ~ ?12 + e12>
where
о°2 = 2цс°12. D6.33)
Then, we have
o[2 = -2^A/2^1212 - l)e{2 + o?2/2S1212. D6.34)
Eshelby's tensor S1212 becomes ^ for a spherical ?2 and Щ for a penny-shaped
J2. The above equation has been obtained by Rudnicki A977 and 1979). a{2
versus e{2 is shown in Fig. 46.4 by the straight line ACDB for a given a°2. If
the constitutive equation of the weakened zone is expressed by curve OCP,

47. Viscoelasticity of composite materials 449
N
Fig 46 4 Stress-strain state inside an inclusion is represented by an intersection of AB and OP (or
06).
point С is uniquely determined for a given o°2 as the intersection of the
straight line and the curve. On the other hand, if the constitutive equation is
expressed by curve ODQ, for instance, the plastic flow inside J2 becomes
unstable, perhaps cracks will be initiated to accommodate deformation.
Rudnicki's theory is used by Rice A979), Rice and Rudnicki A979) in a study
of earthquake premonitory processes. It is argued that the fluid coupling
effects serve to stabilize the weakened rock against rapid fracture, and give rise
instead to a precursory period of accelerating, but initially quasi-static strain-
straining which ultimately leads to dynamic instability (see also Rudnicki 1985 for a
review).
When thin strips are pulled in tension, it is observed that the neck
(shear-band bifurcation mode) forms across the specimen at an oblique angle
which depends on the state of anisotropy. The general theory of bifurcation
has been established by Thomas A961) and Hill A962) and further develop-
developments of the theory have been achieved by Hill and Hutchinson A975), Stbren
and Rice A975), Rudnicki and Rice A975), Needleman A979), Asaro and
Rice A977), and Iwakuma and Nemat-Nasser A982) among others.
47. Viscoelasticity of composite materials
Eshelby A957) has pointed out that the elastic theory of inclusions can be
extended to viscoelastic materials. The viscoelastic creep of composite materi-
materials, for instance, can be evaluated by the viscoelastic theory of inclusions.
Homogeneous inclusions
Consider an infinitely extended viscoelastic material, and let an ellipsoidal
sub-domain Q of the material be subjected to an eigenstrain e*(x, t). The
inclusion J2 and the matrix D — J2 are assumed to have the same viscoelastic

450 Chap. 7 Material properties and related topics
properties. The stress and strain are denoted by otj and с,-_,-, respectively.
Equations B.1), B.2) and B.10) still hold,
€u = eu + erj, D7.1)
where e*¦ = 0 in D — Q. Instead of Hooke's law, we now consider
аи(х, t) = LiJkl{t)ekl{x, 0) + ГLiJkl{t - T) dg^^' T) dr D7.2)
or
etJ{x, t) = Mijkl(t)okl(x, 0) + f~Mukl(t- r)dg^'T) dr D7.3)
assuming that е^(х, t), atj{x, t), LijkI(t) and Mijkl(t) are all zero for t < 0.
LiJkl and Mijkl are called the tensorial relaxation and creep functions,
respectively, and have the same properties of symmetry with respect to the
indices as the elastic moduli and compliances. Let the Laplace transform
(A3.21) of a function f(t) be indicated by a bar,
/(¦s)=f f(t) exp(-st)dt, s>0. D7.4)
•'o
The Laplace transform of D7.1) and D7.2) are obtained as
€ij = etJ + irJ, D7.5)
and
= _ „T p (ли (Л
where
• oo
f M(t-r) exp(-tf) dt = Mexp(-sr)
and
de
I — ехр(-лт) dT= — e@) + se
are used. Equations B.10) for equilibrium are transformed to
*,,,, = 0 D7.7)

47. Viscoelasticity of composite materials 451
and equations B.2) become
c,,= H «,,, + ",,,)• D7.8)
The system of equations D7.5) ~ D7.8) becomes completely equivalent to the
elastic case, where the elastic moduli Cijk! correspond to sLijkl.
For isotropic materials, we have
W) = M0«,-A/ + m(O(M,/ +«//V) D7-9)
and
LlJkl=4j8kl + Д (My/ + Мд ) D7-9.1)
(see Fung 1965). Л and /i in (A2.1) correspond to s\ and jju for the
viscoelasticity. When an ellipsoidal inclusion has a uniform eigenstrain e*7, the
Laplace transform of the stress in Q is obtained from A1.20), by changing ц to
sji and v to v = ^Х/(Х + ft),
1 - 2v f 1 - v v (т Л \-v ]_
D7.10)

452 Chap. 7 Material properties and related topics
and other components are obtained by the cyclic permutation of A, 2, 3). The
inverse transform of a,y is, from (A3.22),
au(x, t) = A/2*0 f+ '°\(*, s) exp(rt) us. D7.11)
The above result can be easily extended to more general eigenstrains and
anisotropic materials. The Laplace transform of the displacement is obtained
from F.2) as
й,= -
Г f sLj^^Ntj/D) «р{/И*-х')} d? dx', D7.12)
*-ooJQ
where N^ and D are the cofactors and the determinant of the matrix having
the ij element sLimjn?m?n. The above equation has been used by Kuo and
Mura A973) for solving a circular twist disclination in viscoelastic materials.
Inhomogeneous inclusions
The equivalent inclusion method in Section 22 can be used for the viscoelastic
creep of composite materials. The constitutive equation of the matrix is
assumed to be D7.2). On the other hand, the inhomogeneities behave as
a°@ + al7(x, 0 =Lfjkl(t){e°kl(O) +<„(*, 0)}
+ Гь*к1A - T)A (?о/(т) + ?fc/(x> T) j dT> D7.13)
where
f\Jkl(t-r)^°kl(r) dr D7.14)
is the applied stress at infinity, and е°_ДО is the corresponding strain. atj in
D7.2) and D7.13) is the stress disturbance due to the inhomogeneities. The
equivalent inclusion method is to simulate ai} by the stress caused by homoge-
homogeneous inclusions with proper ef •. The equivalency is obtained by the equation
D7.13) = D7.14) + D7.2). The Laplace transform of this equation is
sLf^^ll, + ikl) = sLIJkl(i°kl + lkl - Щ D7.15)

47. Viscoelasticity of composite materials 453
which can be obtained directly from B2.5). lkl or a,7 is a linear function of efj.
Equation D7.15) determines Tfj necessary for the equivalent inclusion. The
stress state can be completely determined after e* for a given ajj(t) is
evaluated.
Laws and McLaughlin A978) have estimated the viscoelastic creep compli-
compliances of composite materials and have given numerical solutions for two
common composite materials, namely an isotropic dispersion of spheres, and
an uni-directional fibre reinforced material. Viscous liquids can be treated in
the same manner as in viscoelastic materials if velocity is used in place of
displacement. Bilby, Eshelby and Kundu A975), and Howard and Brierley
A976) have found numerically the change of a viscous inhomogeneity inside a
viscous liquid of different viscosity. Hashin A965) has obtained the macro-
macroscopic viscoelastic heterogeneous media in terms of effective relaxation moduli
and creep compliances. Recently, Budiansky and Hutchinson A980) have
determined the growth-rate and shape for an isolated void growing in a
self-similar way in an infinite block of incompressible power-law, viscous
material (see also Budiansky, Hutchinson and Lutsky 1980). Taya and Seidel
A981), Toon and Taya A984) have obtained the void growth rate when a
viscous material contains many voids. They consider the interaction among the
voids by using Mori-Tanaka's method.
Waves in an infinite medium
Consider an infinite medium which consists of an isotropic matrix and
ellipsoidal inhomogeneities fi with volume fraction с The matrix and the
inhomogeneities are assumed to be different standard linear solids. Setting
D = д/dt, the stress-strain relations are written as
in the matrix D7.16)
and
ta0. D7.П)
where 'atj and 'etj are the reduced stress and strain defined by (A2.6), and
P(D) and Q(D) are polynomials of D.
Consider an applied strain e°y = ё°у ехр(гсог). The strain and stress dis-
disturbances due to the inhomogeneities are also assumed to have the form

454 Chap. 7 Material properties and related topics
tjj = lij exp(/wf) and atj = au exp(iut). If the inhomogeneities are simulated
by inclusions with eigenstrain t*j = I*,¦ exp(iut), the equivalency equations
become
D7.18)
The above equations determine «f, for a given set of e°y since ltj is a linear
function of i*j as shown by A1.15), that is,
itJ = Sijkll%,. D7.19)
Comparing with B2.17), the following correspondence holds:
2,1* = Q* (ш )/P? (/«), K* = Q*2 (ш)/Р{ (ш),
2,i=Q1{m)/Pl(iU), K=Q2(io>)/P(iu) l ' ;
Eshelby's tensor Sijkt in D7.19) contains Poisson's ratio of the matrix, which is
expressed as
p=CK-2ti)/2CK+lx) D7.21)
from (A2.2). К and /x are, of course, functions of P(ioi) and Q(ioi) through
D7.20).
The average (or effective) shear modulus Д and bulk modulus К яте
determined from D5.33) and D5.34), where ju0 = jn, Ko = K, pr = ju*, Kr = K*
and cr = с In the above discussion we have assumed that the state of
deformation is quasi-static. When a dynamic state of the wave propagation
with angular frequency w is considered, we assume that the matenal will
respond by the complex moduli ju and К obtained in the above discussion.
Then, the equations of motion become
V-Kjj + (*Д + K)ujji + P"/ = ° D7-22)
or
о2м
(X + 2Ц)пул - реикект„п„ту + рсо2м,- = 0, D7.22.1)

48. Elastic wave scattering 455
where p is the average density of the material and X + 2JL = К + 4ju/3. The
solutions for D7.22) are the real parts of
щ=Апг ехр(Ц;+ {p/(K+4ji/3)}1/2nkxk}) D7.23)
and
щ = C, ap(ia[t± (p/JiI/2nkxk]), D7.24)
where А, С,, л, are arbitrary constants and Cjtij = 0. Equation D7.23) repre-
represents a plane dilatational wave and equation D7.24) represents a plane shear
wave. The real parts of the factors in front of nkxk are the inverse of wave
velocities, vD, vs; whereas the imaginary parts are attenuation factors aD, as.
We have
vD = \Re{p/(K + 4/1/3) }V2]~\ aD= -o>Im{p/(K+ 4ju/3)}V2,
D7.25)
for dilational waves, and
vs=[Re(p/JiI/2]~\ as=-ccIm(P/]i)l/\ D7.26)
for shear waves.
Walsh A968, 1969) has calculated the wave velocities and attenuations in a
partially melted rock, where the melted zone is simulated by a penny-shaped
inclusion of liquid. The seismic velocities of cracked rocks also are calculated
by Anderson et al. A974), O'Connell and Budiansky A974), and Budiansky
and O'Connell A976), among others.
48. Elastic wave scattering
The subject of multiple scattering of waves is of interest in many fields of
engineering and science. In acoustics, it has important practical applications in
studies of the distribution of flaws in solids, fiber reinforced composites,
porous media, rocks, earth, underwater signal transmissions, etc.
The basic scattering problem is shown in Fig. 48.1. An incident plane wave
with angular frequency w and wave vector к is expressed by
u°(x, t) = u°(x) exp(-iut),
D8.1)
uf(x) = uf exp(ik-x),

456
Chap. 7 Material properties and related topics
Incident
Wave
Fig. 48.1. Scattering of elastic wave due to an inhomogeneity.
the scattered wave by
u',(x, t) = usi(x)exp{-iut)
and the total wave amplitude by
и,(*) = и°(*)+ «?(*)•
The equations of motion are
in the matrix and
= 0
D8.2)
D8.3)
D8.4)
D8.5)
in the inhomogeneity ft, where the elastic moduli and the density in ft are
distinguished by an asterisk, *. Equation D8.5) can be written as
,ij(x) + Рь>2"М + X,(x) = 0,
where
Xt(x) =ACijklukJj(x) +Apo>2Ui(x)
D8.6)
D8.7)

48. Elastic wave scattering 457
and
D8.8)
p* = p + Ap.
The quantity Xj(x) is equivalent to a body force distributed in the domain ?2.
The solution of a unit body force is known as the steady-state elastic wave
Green's function (9.37). Then, for the distributed body force Xt we have
м,.(х) = n,°(x) + jD{ACmjklukJj{x') +Apo>2um(x')}gim(x-x') dx' D8.9)
or
ui(x) = uf(x)+Apcc2fum{x')gim(x-x')dx'
° D8.10)
+ ACm]kljukJ{x')gimJ(x-x') dx'
after integration by parts.
For an isotropic material, gim(x - x') is known from (9.40). The far-field
scattered amplitudes can be calculated by using the following asymptotic
equalities:
1/lx-x'l-l/x,
|x-x'| «x-(x'-x), D8.11)
Э2 exp(/a|x-x'|) «2*<*/
Эх; Эх; i x - x'
where x = x/x.
From (9.40), we have
-exp(/ax - iax' ¦ x),
gu(x - x') = l [р2(80 - x,Xj) exp(if3x - ipx' ¦ x)
4тгрШ2х D8 1
+ a2XjXj ехр(гах — iax' • хЦ ,

458 Chap 7 Material properties and related topics
IX L.
-[j83(S0 - x,Xj) exp(ij8x - ijSx' • 5)
+ a XiX, exp( tax — гад: • x) .
When the expressions in D8.12) are substituted into D8.10), we have
D8.13)
where
/«(«)= -r-~i\APu2(um(x')Qxp(-iax'-x)d.x'
4WPW L Ju D8.14)
t /(дс') ехр(-г'адг' • x) Ax'\
which is called the /-vector. Since мш(х') and uk ,(x') in the integrands of the
integrals in D8.14) are unknown, an iterative procedure is employed. The first
approximation is obtained by substituting u^(x') and ukj(x') for these
unknown functions. This first approximation is called the first Born (or
Rayleigh-Gauss) approximation (see Gubernatis et al., 1977).
The Born approximation becomes progressively worse as changes in the
material properties become large, and is limited to к \ x — x' \ < 1 (long wave-
wavelength). This has been concluded by Gubernatis et al. A977) by comparing the
approximation with the exact solution for a spherical inhomogeneity obtained
by Ying and Truell A956), and Einspruch et al. A960). More convenient
perturbation methods, however, have been developed by Datta A977),
Gubernatis A979), Mai and Knopoff A967), and Gubernatis and Domany
A979), among others.
Another approximation method is based upon the eigenfunction expan-
expansions. According to Morse and Feshbach A958), and Pao and Mow A971), the
dynamic displacement amplitude ut(x) for an isotropic material can be
expressed as
"/(*) = ф,, - "ijkXjt.k - *O/X,, " XjX,i),j D8.15)

48. Elastic wave scattering 459
When it is substituted into the wave equation D7.22.1) (м, for и,-), the wave
equation is satisfied if
D8.16)
are satisfied.
For the spherical coordinates the eigenfunctions of the scalar Helmholtz
equations D8.16) are products of spherical Bessel functions and spherical
harmonics Ytm{8, ф). For non-spherical defects, the expansion method has
been used in several different ways to obtain approximate solutions as seen in
the papers by Waterman A969, 1971), Varatharajulu and Pao A976), Water-
Waterman A976), and Visscher A978).
Recently Gubernatis, Domany, and Krumhansl A980) have published an
excellent review paper on the elastic wave scattering theory with applications
to non-destructive evaluation. For randomly distributed inhomogeneities we
refer to the papers by Bose and Mai A973), McCoy A973), Sobezyk A976),
Datta A977), and Varadan and Varadan A979), among others.
Dynamic equivalent inclusion method
The equivalent inclusion method can also be applied to the dynamic response
of an ellipsoidal inhomogeneity by a slight modification as suggested by Mura
A972c).
The equation of motion is
Cukl{ulu + MM,) = Р(й? + Щ) D8.17)
in the matrix and
C*kl(u°kJj + ukJJ) = p*(u° + fi,.) D8.18)
in the inhomogeneity ?2, where u° is an incident wave and м, is the disturbed
displacement. The equation D8.18) is simulated by the equation of motion in
the homogeneous material with a fictitious body force /, and eigenstrain e? in
Cijkl(ulu + ukJj-eti.j) +fi = p(uf + ui) D8.19)

460 Chap. 7 Material properties and related topics
The equations for the necessary equivalency are
c*jki{»°k,i + uk,i) = Qjki(uk,i + «*,/-«*/). , ,
р*(й,° + «,.) = р(й,0+ «,)-/..
Willis A980) and Fu and Mura A983) used D8.20) for the analysis of dynamic
response of a single ellipsoidal inhomogeneity subjected to plane time-
harmonic waves and evaluated differential cross sections.
Green's formula
Due to the symmetry of Cijkh it holds
j(x~x')uKj(x') = CuklukJ(x')gimJ(x-x'), D8.21)
where g,7(* - *') is defined by (9.36). The formula resulted from integrating
D8.21) with respect to x' over any domain is called Green's formula. Green's
formula has been used in the boundary integral equation (BIE) method
developed by Kupradze A953, 1965). Using this method, Kobayashi and
Nishimura A982) Niwa, Kitahara and Ikeda A984), Kitahara A985) among
others have established numerical schemes to solve transient wave propagation
problems around inhomogeneities.
Let us apply the BIE method to the problem shown by Fig. 48.1. Integrate
D8.21) for the scattered wave amplitude (take м, as u*) in the matrix domain
(D — Q), and apply integration by parts. Then, we have
-f Сиклкт,,(х-х')и',(х')пу dS(x')
fD
D_aCijklgkm,lj(x-x')u't(x')dx'
Cuk,ulu{x')glm(x - x') d*\ D8.22)
where «7 is the outward unit normal on the boundary | i21 of the domain ?2.

48. Elastic wave scattering 461
Since the scattered wave vanishes at infinity, the boundary integrals at infinity
become zero (the radiation condition).
The relations CiklgkmJj(x -x')= - pu2gim{x - x') - 8,J(x - x') and
Cjjkluklj{x') = -роУ-и]{х') are substituted in D8.22) to yield
f 8(x-x')u'Jx')dx'=-f CiJklgkmj(x-x')uttx')njdS(x')
JD-U J\ii\
D8.23)
C,jklukJ{x')gim{x- x')nj &S{x').
The left hand side in the above Green's formula is usm{x) when x is in the
matrix D — п, \usm{x) when x is on \Q\, and zero when л: is in Q, because of
the definition of Dirac's delta function.
When the incident wave amplitude uf is used for м, in D8.21) and the
identity is integrated in the domain $2, we have similarly
[8(x-x')u°m(x')dx'= f CuklgkmJ(x-x')uf(x')njdS(x')
D8.24)
+ / Сик1и1,(х')ьт(х-х')п; dS(x'),
since Cljklu°klj(x')= -pu2u?(x') in i2 and D - п. When the formula usm =
um ~ um is substituted into D8.23) and relation D8.24) is used, we have
f S(x-x')uJx')dx'=-[ Сик18кт1(х-х')и,(х')п^Б(х')
C,jklukJ{x')gim{x-x')nj&S{x') D8.25)
\
fs(x-x')u°m(x')dx'.
JD
Iii order to apply D8.21) to и, defined by D8.5), a new steady-state elastic
wave Green's function gkm(x — x') is defined to a homogeneous infinite

462 Chap. 7 Material properties and related topics
domain with the elastic moduli C*k/ and the density p*,
C*jkigLjj (x - x') + P*<*2gL (x - x') + Sim8 (*-*') = 0 D8.26)
The Qy-fc, and gA:m in D8.21) are replaced by С*ы and g*m(x - x') and the
resulting identity is integrated in domain ft with respect to x'. Then, an
expression similar to D9.24) is obtained,
J8(x - х')ии(х') dx' = f C*Jk,gtmJ(x - x')u,{x')nj dS(x')
|Й| D8.26.1)
The left hand side of D8.26) is um{x) when x is in Q, \um{x) when л: on | Q \
and zero when x is in D — i2.
The boundary integral equations are obtained from D8.25) and D8.26) by
taking x on \ii\ and by requiring the continuity conditions for the displace-
displacement and the force tractin on \Q\. Writing the force traction as tt =
Ciikiuk,inj = C*jkiukjrij, we have for x on | ft |
*«m(x) = - / C^g^X-x'Mx')^ dS(x')
W D8.27)
J\a\ '
and
Ах-х')и,(х')^ dS(x')
D8.28)
+ Г r;(x')g;*m(x-x')dS(x')
•'ini
The above equations D8.27) and D8.28) are the integral equations to de-
determine м, and tj on \?2\. After determination of these boundary values, the
displacement um in D — Q is evaluated by D8.25), and um in п by D8.26),
where ?,(*') is used for CuklukJ(x')nj in D8.25) and C*jklukl(x')nj in D8.26).

49. Interaction between dislocations and inclusions 463
When the material contains many inhomogeneities, пъ fl2, ... we have the
same boundary integral equations D8.27) and D8.28) by taking | ?2 \ =
| flj | + | fl2 I + When the material is a half space, still the same equations
hold if the other half space is taken as an infinity large void ?20, and
|O| = |O0| + |O1| + |O2| + ....
When an incident wave is transient, it is expressed by the Fourier integral
form with respect to u. Then, the transient solution is obtained by superposing
the steady-state solutions described here.
49. Interaction between dislocations and inclusions
The term inclusion is interpreted here in a broader sense to include inhomo-
geneity and crack. Dispersed second-phase particles and nonmetallic inclu-
inclusions exert a large influence upon the mechanical properties of materials. In
addition to increasing the yield strength by raising the stress necessary to move
dislocations through the matrix, the presence of dispersed phases and inclu-
inclusions affects the fracture behavior of these materials by providing sites for
crack formation via particle or particle-interface cracking at the tip of blocked
slip bands.
Inclusions and dislocations
Consider an infinite elastic medium containing a circular cylindrical inclusion
of radius a (Fig. 49.1). The shear modulus and Poisson's ratio of the matrix
are denoted by jux and vx and the corresponding constants of the inclusion by
ju2 and v2. The stress field of a dislocation located at (?, 0) is found by
Dundurs A967) for a screw dislocation and by Dundurs and Mura A964) and
Dundurs and Sendeckyj A965) for an edge dislocation. The evaluation of the
Peach-Koeler force requires the stress component ayx for the screw dislocation
and ayz for the edge dislocation.
For the screw dislocation, the stress at (jc, 0) is
where x > a.

464
Chap. 7 Material properties and related topics
Fig. 49.1 A dislocation near a cylindrical inhomogeneity
For the edge dislocation, the stress at (x, 0) becomes, for x > a,
A+B
2(x-a2/?)
-А^У-'п-
i/a x-a1
аЛ
/1}
(x
-a2 A)"
D9.2)
where b is the Burgers vector, and
A = A -
В=(к2-
D9.3)
= 3 ~ 4v2.
Matsuoka et al. A976) and Saito et al. A977) have considered the effect of
the free surface located near to the circular inclusion and a dislocation, more
complicated cases have been considered by several authors. When a circular
inclusion has a slipping interface, the elastic interaction between an edge
dislocation and the inclusion is investigated by Dundurs and Gangadharan

49. Interaction between dislocations and inclusions
у
/
465
О/
—c-H
Fig. 49.2. A dislocation near a two-phase material.
A969). The elastic interaction between a circular inclusion in a semi-infinite
body and a screw dislocation is given by Tamate A973). Sendeckyj A970) has
given the elastic field of a screw dislocation near an arbitrary number of
circular inclusions. Tamate A968) has investigated the behavior of a screw
dislocation near a partially bonded bimetallic interface. The case for an edge
dislocation is considered by Tamate and Kurihara A970). A dislocation near
or on a surface layer is treated by Weeks et al. A968), Lee and Dundurs
A973), Masumura and Ghcksman A977) for the edge case. Gavazza and
Barnett A974), and Willis et al. A972) have found the general solution for the
interaction between a spherical inclusion and a screw dislocation. If the
inclusion and a dislocation are far apart, the long-range interaction approxi-
approximation is possible as shown by Barnett A971), Comninou and Dundurs
A972), Yoo and Ohr A972), Lin and Mura A973), Yoo A974), and Heinisch
and Sines A976). The interaction between a crack and a dislocation has been
investigated by Tamate and Sekine A972), and Rice and Thomson A974).
Hasebe et al. A984) calculated the stress intensity factor of a kinked crack
initiating from a rigid line inclusion.
If the inclusion is a half-space, we take ?/a -» 1, {(?/aJ — \}a -» 2?,
a - аг/? -»i and x - a2/? -> x + ? in D9.1) and D9.2) (see Fig. 49.2). Then,
D9.1) and D9.2) become
~ \
2m \ x -
D9.4)

466 Chap. 7 Material properties and related topics
for the screw dislocation, and
for the edge dislocation.
The above solutions agree with those obtained by Head A953). The
Peach-Koehler force can be obtained by azyb or ayxb at x = ?, where the term
l/(x -1) is omitted for the calculation.
The problem of a linear array of length L containing n dislocations piled
up along the x-axis against the circular inclusion under the application of an
applied stress az°y (for screw) or ayx (for edge) may be formulated as follows.
For static equilibrium the force acting on any one dislocation in the pile-up
due to all other pile-up dislocations must balance the force due to the applied
stress. This requires that azy + az°y = 0 or ayx + ayx = 0,
D9.6)
for i = 1, 2,..., n — 1.
for the screw dislocation array. A similar equation holds for the edge disloca-
dislocation array.
When n is large, the discrete dislocations in the array can be replaced by a
continuous distribution of dislocations with density /(?)• Equation D9.6)
becomes
for a < x ^L.
Solving D9.6) yields values of xx, x2,..., xn_x and n for a given constant
az°y. Equation D9.7) determines f(x) and L for a given az^.
Solutions of these equations are obtained by many researchers (see a review
paper by Chou and Li 1969). Eshelby, Frank and Nabarro A951) have solved

49. Interaction between dislocations and inclusions 467
the discrete dislocations pile-up problem in the homogeneous medium (ju2 =
Mi. "i = )»
n-1 -
I
XJ
-0 = 0 fori = l,2,...,ii-l, D9.8)
where
ayz>
for the screw dislocation case, and
"l ' D9.10)
°= ~
for the edge dislocation case. Equilibrium equation D9.8) holds for the (n — 1)
locations. The leading dislocation at x0 is a locked dislocation. Without losing
generality, we can assume that x0 = 0 and an obstacle is locking the leading
dislocation. Eshelby et al. obtained хг, x2,...,xn_1 as the roots of the
generalized Laguerre polynomial L^l^x) by taking D/2a as a unit of length.
The stress atip acting on the locked dislocation is obtained as (n — l)a, from
the virtual work argument of Cottrell A949). When the leading dislocation is
displaced by 8x0, the other (и - 1) dislocations are also displaced by 8x0
without losing the equilibrium since only the relative location of the disloca-
dislocations is essential for the equilibrium. The work done by the applied stress a on
the (и - 1) dislocations during the virtual displacement Sx0 is aft (и — l)Sx0
which must be equal to the work done by the locking force atipb on the leading
dislocation. Namely, ab(n — V)8xQ = аирЬ8х0, which leads to atip = a(n — 1).
Stroh A954, 1955) and Y.T. Chou A967) have calculated details of the
stress field and energy associated with the Eshelby, Frank and Nabarro
problem. A modification of the Eshelby, Frank and Nabarro problem is made
by Head and Thomson A962) by considering a variation of the applied stress
and by Y.T. Chou A967) by taking a variation of the strength of the locked
dislocation. J.C.M. Li A969) has investigated tht case of extended dislocations.
The discrete dislocations in the Eshelby, Frank and Nabarro problem can also
be replaced by a continuous distribution of dislocations as done by Leibfried

468
Chap. 7 Material properties and related topics
Fig. 49.3. Pile-up dislocations at the interface.
A951). He has solved the integral equation
-o = 0 forO<Jc<L,
and has obtained
The total number of dislocations is found to be
„= (Lf
D9.11)
D9.12)
D9.13)
in agreement with the discrete dislocation case.
The pile-up problem in a two-phase medium (two half-spaces) requires
solving an integral equation of the type (see Fig. 49.3)
D9.14)
or
ln-1
л-1
¦х гi — i ri — j
— - E -JL-~ E
for i= 1, 2,..., n - 1,
*, 7Гох' + ху y-o
D9.15)

49. Interaction between dislocations and inclusions 469
where definitions D9.9) and D9.10) are used, and
a=(til-ti2)/(n1 + [i2), P = 0 (for screw),
D9 16)
a=\(A + B), P = 2A (for edge).
The solution of D9.14) is found by Y.T. Chou A965), Bamet and Tetelman
A966), Barnett A967), and Smith A967) for the screw dislocation case, and by
Kuang and Mura A968) for the edge dislocation case. The distribution density
of the screw dislocations is
/(*)= 71—rsinhb cosh-HV*)}, O^x^L, D9.17)
ftjU sinews)
where
s = BA)sin-1{j"i/(Mi + M2)}1/2- D9.18)
The number of screw dislocations in the pile-up is then found to be
n = (aLs/D)/sm(s4r). D9.19)
The stress at the tip of the pile-up becomes
D9.20)
where r is the distance from the tip in the x-direction. The order of stress
singularity at r = 0 depends on the shear moduli of the two phases. The value
of 5 is 1/2 for the homogeneous (jux = ц2) case.
The corresponding quantities for the case of edge dislocation pileup are too
complicated to record here. Kuang and Mura A968) showed that these
quantities are depending on the roots s of the equation
cos(irs) + a-/3(s-lf = 0. D9.21)
Numerical examples of the edge dislocation distributions are shown in Fig.
49.4 where Г = /x2/jWi- These distributions are very close to the screw disloca-
dislocation distributions given by D9.17).

470
Chap. 7 Material properties and related topics
Q
b
08
0 7
0.6
05
04
03
0 2
0.1
n
¦ \\\ V
\\W\a
¦ \
Г /3 —
Г" 7
Г» со —
¦
— г-
г-Г •
\
1/7
3/5
1
\
0.2 0.4 06 08 10
C/L
Fig. 49.4. Distribution of pile-up edge dislocations.
The nature of the stress singularity at the tip of the edge dislocation pile-up
can be investigated by the following alternative method. Equation D9.14) is
replaced by
"Пт-t-Th-
JQ \x-? x + $
forO<(x/L)<e,
D9.22)
where € «: 1 and H(x) is nonsingular as x -> 0+.
Assuming that f{i)v& of the form
k (O<A:<1), D9.23)
we get
ftftDxk~x{cos{"nk) -a + 0k2 }/sin(irA:) - a + H(x) = 0, D9.24)
where H(x) is not singular as x -* 0+. Since only the first term in D9.24) is

49. Interaction between dislocations and inclusions 471
singular as x -»0+, for identity to hold, the following relation must be
satisfied:
-cos(wA:) + a-/?A:2 = 0. D9.25)
The above equation is equivalent to D9.21) when s — l = k. The stress at the
tip of the pile-up can be evaluated from the same integral in D9.22) except
x < 0. The integral gives
f /o / 2k + 12k-1 \ л.к
°n Ma sin(ir*)\l+ C-4
D9.26)
where r= \x\. The order of stress singularity at the tip is the same as that of
dislocation distribution D9.25). Equations D9.22) ~ D9.24) are presented by
Keer A968) in a private communication.
Barnett A969) has shown that
0<(\-k)<\ if ^/^ C-4^OC-4^),
4)
The relation between к and Dundurs constants defined by F.15) is discussed
by Dundurs and Lee A970).
The problem of screw dislocation pile-up against a circular cylinder requires
solving D9.7). The exact solution is given by Barnett and Tetelman A966,
1967). Toya A976) has investigated the interfacial debonding caused by a
screw dislocation pile-up at a circular cylindrical rigid inclusion.
Cracks in two-phase materials
The equilibrium conditions for pile-up dislocations can also be interpreted as
the conditions for a fracture when the conditions for the free (crack) surface
are identical to the equilibrium conditions for the dislocation array under an
applied stress. Tucker A973) has used the method of Kuang and Mura A968,
1969) (the Melin transform and the Wiener-Hopf technique) and analyzed the
behavior of Griffith, Zener-Stroh and Bullough-Gilman cracks in bi-metals.
The effect of a circular inclusion on the stresses around a slit-like crack is
investigated by Tamate A968).
When a crack is located along the interface of a bi-metal, the well-known
elastic solutions (e.g. Williams 1959, Erdogan 1963, England 1965, Rice and

472
Chap. 7 Material properties and related topics
И-г- "г
Fig. 49.5. A crack at an interface of a two-phase material.
Sih 1965, Clements 1971) characteristically involve oscillatory singularities.
Comninou A977) has reconsidered the problem using a continuous distribu-
distribution of crack dislocations and found a non-oscillatory solution. She assumed
that the crack is not completely open and that its faces are in frictionless
contact near the crack tips (see also Comninou 1978, Dundurs and Comninou
1979, Comninou and Schmueser 1979). It has been shown by Achenbach et al.
A979) that the introduction of a small cohesive zone of a certain characteristic
in the vicinity of the crack tip also serves to remove the corresponding
oscillatory singularity. Recently, Hayashi and Nemat-Nasser A981) have
considered a branched crack at the interface of two dissimilar materials.
The analysis of Comninou is as follows. Consider a crack of length 1L
lying in the interface of two elastic solids with shear moduli /x,j and ju2 and
Poisson's ratio vx, v2 as shown in Fig. 49.5. Under the action of uniform
tension T applied at infinity, in the direction normal to the interface, the crack
opens in the interval { — a, a), where a is an unknown to be determined in the
course of solution. We assume that a portion of the crack is closed and its two
sides are in frictionless contact in the intervals { — L, —a) and (a, L). The
boundary conditions are that the shear traction must vanish in ( — L, L) and
that the normal traction must vanish in (-a, a). To enforce these conditions
we take advantage of the known stress fields for edge dislocations and
consider a distribution Bx(x) of glide dislocations in the interval ( — L, L),
and a distribution By(x) of climb dislocations in the interval ( — a, a).

49. Interaction between dislocations and inclusions
When a discrete edge dislocation is located at point ? on the interface
between two solids, the induced interface tractions are (see Comninou 1977)
oyy(x,0)=-BCbx8(x-0, D9'28)
for a glide dislocation, and
D9-29)
Oyy\X, \}) = K,Uy/TT\X — ?),
for a climb dislocation, where S(x — ?) is the Dirac delta function, and bx and
by are the components of the Burgers vector. Moreover,
2M2(l + «) _ 2Ma(l«)
( + l)(l/?2) ( + l)(l82)' l • '
where a and В are the Dundurs constants F.15).
The boundary conditions on the traction mentioned before lead to the
integral equations which determine Bx and By. They are
? = 0, -L<x<L,
( Щ
7Г J^L X — ?
D9.31)
=0, -a<x<a, D9.32)
where H(x) is the Heaviside unit step function. The condition for single-val-
single-valued displacements for \x\ > L requires that
4({)d{ = 0, D9.33)
By(?) dt = O. D9.34)
-a
Denoting by g(x) the gap between the solids and by h(x) their relative slip,
we have
( • '

474 Chap. 7 Material properties and related topics
The definition of continuous distribution of dislocations C8.8) gives
Bx(x) = -dh/dx, By(x) = -dg/dx. D9.36)
Thus, we can assume that By(x) is continuous in ( — a, a) and bounded at the
ends -a, a. Then, D9.31) can be viewed as the Hilbert integral equation
(A4.1) for the unknown function Bx(x) and solved formally by treating By(x)
as known. Assuming Bx(x) is unbounded at —L and L, as the asymptotic
analysis (Comninou 1977) indicates, Bx(x) is obtained as (A.4.3), where
ax = -L, bx = L, c1+q = —L, c2+q = L, p = l, q = 0. The solution, using our
condition D9.33), becomes
D937)
where
x = x/L, i/L=t, a/L = a. D9.38)
Condition D9.34) is satisfied by symmetry. Since By is bounded, Bx(x) is
square root singular at L and — L.
Substituting Bx(x) from D9.37) into D9.32), we have
o2 n (л ЛЧ1/2
-a<x<a
D9.39)
which is a generalized Cauchy singular equation to determine By(t). A
numerical solution can be obtained by the method developed by Erdogan and
Gupta A972).
The singular behavior of the interface tractions can easily be established.
The 'ractions at the interface are
sgnx P BAt){\ - t2f/2
Sgnx f уУ А 7 ^ _>;l> D94Q)
(x2-i
-
t-X

49. Interaction between dislocations and inclusions 475
and
^f ^Щйй, \x\>L
-^jdZ, L>\x\>a. D9.41)
It can be seen that oxy{x, 0) has square-root singularities at L and — L and
ayy(x, 0) is bounded at L+, —L+ but has square-root singularities at L~,
-L~ because of the term - CfiBx{x). Recently, Мак et al. A980) have shown
that both the normal and the shear stresses are singular at the crack tip when
the interface adhesive zones have no-slip.
Finally, the solution of D9.39) must satisfy the following conditions: for no
interpenetration,
g(x)>0, |x|<a, D9.42)
and for contact,
a
yy
(x,0)<0, a<\x\<L. D9.42.1)
Comninou showed a numerical example for the case a/L = 1 — 10 ~4 and
/3 = 0.4854. Her result shows that axy(x, 0)(x2/L2 - \)l/1/T = 1.050,
ayy(x,0)/T= 23.36 at x/L = 1.0000. These values cannot be obtained from
the classical solution (e.g. England 1965) due to its oscillatory singularities.
However, her solution is very close to England's solution when x/L > 1.0002.
England's solution is
TLl/2
oxy(x,0)=——~—-Be costs-sines), x>L, D9.43)
[2(x — L)\
where
?=2^TTf '-«log^. D9-44)
The crack extension force 48X defined by C4.6) becomes almost identical in
both of the solutions and it can be approximated by
D9.45)

476 Chap. 7 Material properties and related topics
where
АГ2 = Г1У2A + 4е2I/2, ?*0. D9.46)
Comninou's consideration has been extended to a penny-shaped crack at
the interface of two bonded dissimilar half-spaces by Keer et al. A978). By
considering contact zones, these solutions have no oscillatory singularities. The
solutions obtained by Mossakovskii and Rykba A964), Keer A967), Willis
A972), and Lowengrub and Sneddon A974) have the oscillatory singularities
since their penny-shaped cracks are of Griffith type. However, for cracks in
tension fields the oscillatory zone is very small compared to the radius of the
crack and these solutions give good approximations outside the zone. The
oscillatory zone is no longer small for a crack under shear fields. The solution
of Lowengrub and Sneddon A974) for a crack under internal pressure p gives
D9.47)
at the interface z = 0, where p is the dimensionless radius,
D9.48)
and
о
l
S2(p)=-1A f1 *rin(*> d*. D9.49)

50. Eigenstrains in lattice theory 411
50. Eigenstrains in lattice theory
In lattice theory (e.g. Born and Huang 1954, Maradudin 1958) a change of
potential energ) caused by a deformation is expressed by a function of atomic
displacements measured from a perfect lattice state. For harmonic approxima-
approximation the potential energy is expressed by a quadratic form of the atomic
displacement components. The coefficients of the polynomial are chosen such
that a uniform displacement (rigid motion) does not create any potential
energy of deformation. The potential energy, however, does not vanish when a
part of the material is displaced plastically and reformed into a perfect crystal.
When a perfect crystal is deformed into another perfect crystal accompanying
only a plastic shape change, no change of potential energy is expected.
Therefore the classical lattice theory should be modified so that the potential
energy does vanish for such a plastic deformation. A similar question about
the incompleteness of the classical theory has been raised by Kuriyama A967)
from a different viewpoint.
It is shown that such a difficulty in the classical lattice theory simply can be
eliminated when the concept of eigenstrains in continuum mechanics is
introduced into the lattice theory. In continuum mechanics the eigenstrain has
played an important role in analysis of elastic fields caused by line and point
imperfections, mclusions, inhomogeneities, etc. The introduction of the con-
concept of eigenstrains into lattice theory provides similar advantages and
eliminates the above-mentioned difficulty in classical lattice theory.
The equation of motion of atom / in the classical lattice theory with the
harmonic approximation is
Мм,.(/)=-2>,7(Я')и,(/'), E0.1)
v
where the summation with respect to /' is referred to all atoms, M is the mass
of an atom, ut{l) the displacement component from the perfect lattice state of
atom /, indicating the position, and Ф,-7(//') are the force constants between /
and /' atoms. The double indices appearing in the equations follow the
summation convention from 1 to 3. The force constants have the following
properties, since the right-hand side in equation E0.1) is zero for a constant
displacement,
?*,;(//') =ЕФ„(//')=О. E0.2)
Furthermore, the symmetry condition leads to
E0.3)

478 Chap. 7 Material properties and related topics
The summations with respect to /' in E0.1) and E0.2) are taken for all atoms
including /. It is obvious from E0.2) that Егф,7(//')м,(/) = О. Then E0.1) can
be written as
E0-4)
If we put
E0.5)
E0.6)
where xk(l) is a coordinate component of / atom in the perfect lattice state,
we write E0.4)
E0.7)
It is important to keep in mind that E0.1) and E0.7) are equivalent only
when Pkj(W) is compatible, namely, only when E0.5) holds. However, E0.5)
does not always hold. If the material is involved in any inelastic phenomena,
for instance, in plastic deformation, E0.5) does not hold and instead
«,(/') - «,(/) = [pkJ(W) + ftj(U')]xk(n). E0.8)
Now we can call fikJ and /?*y the elastic distortion and eigendistortion,
respectively, and their symmetric parts are called the elastic strain and
eigenstrain, respectively. These definitions are strictly analogous to those of
continuum mechanics, and the total strain is the sum of the elastic strain and
eigenstrain. We assert that the fundamental equation of motion is E0.7) rather
than E0.1), and condition E0.8) is subjected to equation E0.7). Equation
E0.7) becomes E0.1) only when /??y = 0. The plastic deformation which
transforms a perfect crystal into another perfect crystal, accompanying a
permanent shape change, can be described by zero elastic distortion (or
strain), that is, fikJ = 0. In most inelastic problems ы, and fik] are unknown
quantities to be determined, while the components of fi*j are a priori given by
the nature of the problems. For plastic deformations, fikj(H')xk(l'l) and
ft*j(H')xic(l'l) are tne elastic and plastic displacements, respectively. The last
displacement corresponds to the magnitude of glide which is a multiple of the
lattice spacing. Analytically, however, it is expressed by a continuous function

50. Eigenstrains in lattice theory 479
of coordinates of a generalized function. /ikj(ll') and P*j(ll') are defined
between atoms / and /' and are equal to (}kj(l'l) and /??Д/7), respectively.
When Pkj{W) is eliminated from E0.7) and E0.8), we We
E0.9)
Comparing E0.9) and E0.1) it can be seen that an extra term appears as a
fictitious body force acting on atom /.
The potential energy in the classical lattice theory is within the harmonic
approximation
@и/О E0.10)
or
E0.10.1)
The last expression E0.10.1) has been obtained by the use of E0.2) and the
equality
E0.10.2)
where the summations E^-i and Ei-д are combinational and permutational,
respectively.
Our modified theory asserts that the potential energy is not associated with
the total displacement м, but the elastic displacement defined by f$ki{ll')xk(l'l).
Then, the new expression of the potential energy becomes, instead of E0.10),
Ф = - \ E Фи{11')^{11')хк(П)^A1')хт(П). E0.11)
If a component of stress tensor is defined as
E0.12)
E0.11) can be written as
Ф=*Ь3 ?*,*(//')&,¦(»')¦ E0.13)

480 Chap. 7 Material properties and related topics
The definition of stress E0.12) has been chosen such that it agrees with one in
the continuum mechanics for limiting case b-*Q. Similarly to continuum
mechanics, the potential energy (the elastic strain energy caused by /}?,) can be
written as
ф=-*а3 !>/*('/')&•,(//') E0.14)
г-/
for the static case. It is proved as follows. Equation E0.11) is written from
E0.8) as
Ф= -* I ф,7(//')^(//')хт(/7)[М,(/')-«,(/)-)8,*.(//')^(/7)]. E0.15)
r-i
For the static case E0.7) and E0.3) lead to
= о,
and therefore E0.15) can be simplified to
E0-17)
which is equivalent to E0.14) due to E0.12). Equation E0.14) has the same
form as A3.3).
A uniformly moving screw dislocation
A straight screw dislocation along the хъ axis is uniformly moving to the хг
direction with a constant velocity v. At time t = 0, /i/2,i/2> A/2-1/2 atoms (see
Figure 50.1) are in a quarter way jumping the next potential well. For
mathematical simplicity, only the nearest neighborhood interaction is consid-
considered for a cubic crystal. Now we assume that
Jm,n+\)= -В, E0.18)
JmJ = 2(A+B).
since the plastic displacement /?2з(//')х2(/7) must be b, we take /?2*3(//') = 1

50. Eigenstrains in lattice theory
481
о
о
о
о
о
0
о
о
О-к
ь
6
о
0
0
0
о
-о
о
о
о
0
о
о
о
о
о
о
о
о
Х2
о
о
о
о
о
о
о
0
0
0
vt
о
о
О
о
О
о
о
О
о
о
о
О
о
о
J
О
о
о
О
о
о
о
о
О
о
о
Fig. 501 A uniformly moving screw dislocation with velocity v. The atom positions are denoted
by integers /, m and я
between / = lk+i/2,i/i and ''= ^+1/2,-1/2 atoms (k = integer between —00
and vt/b). Then, our fundamental equation E0.9) becomes
M«3(/m>J =A[u3(lm+hn) + и3
+ В[и3Aт<я+1) +
vt/b
+ ВЬ X (^
,n-i)] - 2(A + В)и3Aт,я) E0.19)
where 8m „ is the Kronecker delta. Although vt/b has been assumed as an
integer, the analytic nature of our final solution can extend the solution to any
value of vt/b.
Since
E0.20)

482 Chap. 7 Material properties and related topics
the solution of E0.19) is assumed as
\ r™ />_ { t Vt \ \
E0.21)
(^1. ?2) is easily obtained as a solution of an algebraic equation when
E0.20) and E0.21) are substituted into E0.19). Then, we have
. . b b fn/2 r sin ?2
M3V m,n) 4 „2 Jo J Sin ?j
E0.22)
where
X=2(m-vt/b), V2 = Mv2/Bb2 = v2/c2, E0.23)
and с is the sound velocity of shear waves. The plus and minus signs of the
first term in the right-hand side in E0.22) are taken for n > 0 and n < 0,
respectively. When V = 0, E0.22) agrees with the result obtained by Maradu-
din A958). When A = В and ^/b = 1Ъ i2/b = |2, mb = хъ nb = x2, b-> 0,
E0.22) leads to the classical solution of continuum theory,
"з(*1> х2) = {Ь/2чтIа.п-1(х2/х'1), х[ = (хг - vt)/{\ - V2I/2. E0.24)
The displacement due to applied stress of23 = a is obtained as
"з('т,л) = nb2a/B. E0.25)
The total displacement w is
w= E0.22)+ E0.25). E0.26)
The displacement field is anti-symmetric about x2 = 0 plane. The displace-
displacement of atoms just above the slip plane is obtained by taking n = \. The
numerical values are shown in Figure 50.2 for various values of V in case
A/B = 1 and В = fib (fi = shear modulus), where a/ju is chosen such that
w/(\b) = 0.25 at X= 1 E0.27)

50. Eigenstrains in lattice theory
483
Fig. 50.2. The displacement of atoms just above the slip plane for various velocities of the
dislocation, where V = v/c The dislocation is in the position X = 1.
is satisfied. The last condition means that the atoms at the dislocation position
are in a quarter way jumping to the next potential well. The applied stress a
satisfying condition E0.27) becomes a function of V as shown in Figure 50.3.
@ 100
(СГ//X)x|03
Fig. 50.3. Applied stress necessary to keep the dislocation in motion with velocity v.

484 Chap. 7 Material properties and related topics
In continuum theory no applied stress is necessary to maintain the disloca-
dislocation in the uniform motion for V < 1 and the displacement profile of the upper
slip plane is w/\b=\ for X < 0 and w = 0 for 0 < X.
Comparing the result shown in Figure 50.2 with the result of one-dimen-
one-dimensional chain model obtained by Earmme and Weiner A974), the present
theory predicts the damping of oscillation of atoms in the trail of the
dislocation and also predicts an oscillation of atoms ahead of the dislocation.
The theory in this section can be applied equally to vacancies and moving
cracks as shown by Mura A978).
There seems to exist a modern trend that the micromechanics goes to more
microscopic scales as seen in recent publications, e.g. Kunin A982), Weiner
A983), Yonezawa and Ninumiya A983), Monnaga et al. A984) and Eberhart
et al., A985).
51. Sliding inclusions
Inclusions and inhomogeneities considered hitherto are perfectly bonded to
matrices. The condition of perfect bonding is sometimes inadequate in describ-
describing mechanical behavior of inclusions. Inclusions in high-strength steel, for
instance, are easily debonded by a few cyclic loadings. Grain boundary sliding
in polycrystals and granular media, debonding of fiber elements in composite
materials are also common phenomena.
Consider an infinitely extended elastic body D, containing a uniform
eigenstrain ef • in an ellipsoidal subdomain Q. We investigate a solution of the
elastic field when sliding takes place along the inclusion interface S. Although
some laws of friction must be introduced on the inclusion interface, it is
assumed, for simplicity, that the interface cannot sustain any shear traction.
This free sliding inclusion is subjected to the following conditions:
a,jj = 0 inD
аи = Cijki"k.i in D - 12
[ojj]rij = 0 on S
[м,]и, = 0 on S
aij"i ~ °)кП}пкп, = 0, on S
where
[ ] = [out] — [in], and и, is the outward normal to S.

57. Sliding inclusions 485
The last two conditions are different from the conditions for the perfect
bonding. For the perfect bonding, [ы,] = 0 and the shear stress along the
interface, a^rij - ^кпупкпп has no restriction.
The shear stress on the inclusion interface in the perfect bonding case
(Eshelby's inclusion case) is relaxed by the interface sliding (Somigliana's
dislocation creation). Therefore, the solution for E1.1) is the sum of Eshelby's
solution F.1) and Volterra's solution G.6) with Somigliana's dislocations bi
defined by
-[«,] = bt. E1.2)
The bj must be determined from the last two conditions in E1.1). Then, the
displacement is
и, = и,* + иГ, E1.3)
where
E1-4)
The corresponding stress is
a,7 = a*+<, E1.6)
where
v r v у?1-1)
atj = 4jkiuk,i-
Following Asaro A975), b defined on 5 is extended to the inside Q. Then,
Gauss' theorem is applied to E1.5),
E1.8)

486 Chap. 7 Material properties and related topics
Therefore, we have
a/j(x) = -CiJkl{ j Cpqmrli*nm{x')Gkp,ql{x-x') ux' + e*kl
E1.9)
where
e*;=-|(^ + ^) E1.10)
Shearing Eigenstrains
Mura and Furuhashi A984) have found that the stress a,; in E1.1) is
identically zero when tff is of the shear type, ef2, cf3, c^i» where the
coordinate system is taken in the principal axis directions of ?2.
Observing E1.9), we realize that a,7 = 0 if
e?, + c?/ = 0. E1.11)
and all the conditions in E1.1) are satisfied except the condition [м,]и, = 0 on
S. It is known that [м,] = — bt. The condition
b,nt = 0 onS E1.12)
is, however, satisfied when e* is of the shear type. Because E1.10) and E1.11)
lead to
fe, = ***,.-со;Л E1.13)
and ии= —oijj is a rotation which is chosen so that the condition E1.12) is
satisfied. Since c? is of the shear type, when E1.13) is substituted into E1.12),
we have
{ef2(l/fl? + l/a\) - ul2(l/al - l/a22)}Xlx2
E1.14)
= 0

51. Sliding inclusions 487
where
«2 = (*г/а1)/{х?/а{ + x\/a\ + х\/а\)Х/\ E1.15)
«3 = (*э/«
When ахФ агФ а3, there exists a set of solutions of «,y satisfying E1.14).
They are
- 1A22),
o31 = €^(l/a3 + l/a)/(l/a3 - V«i2), E1.15)
«23 = f?3(V«2 + 1/в|)/(l/fll - 1/ef )•
Therefore, it is concluded that for uniform shear eigenstrains a free-sliding
ellipsoidal inclusion causes no stress field. This means that the misfit caused
by e*j in Eshelby's inclusion is completely relaxed by the interface sliding
given by E1.13) with E1.15). It means equivalently that the displacement
u, = efjXj - uuXj transforms the ellipsoid x\/a\ + x\/a\ + x\/a\ = 1 into the
identical ellipsoid {x\J/a\ + (x'2J/al + (x'3J/aj = 1 by the transformation
*; = *, + ?>,. E1.16)
This is surprisingly true in the framework of small deformation theory.
When Q is a sphere (аг = аг = аъ = a), the terms in E1.14) containing co,y
disappear and therefore bt expressed by E1.13) does not satisfy the condition
bini = 0. Mura and Furuhashi A984) have found that when
bt = BijXia2 - Bjkx}xkXi, E1.17)
with
14G-5»)
3B5 -2v)a2
"и-^НЪ'Ь (ял.)
is chosen, ut and atJ evaluated from E1.14) E1.15), E1.9), E1.10), satisfy all
the conditions in E1.1). v is Poisson's ratio and a is the radius of U. The
expression E1.17) is found from the intuition that bt must have the same form
as the shear stress, o^XjO2 — OjkXjXkxt, at the interface. It is also suspected

488
Chap. 7 Material properties and related topics
from the solution of Ghahremani A980) who has obtained the solution for an
isotropic elastic medium, containing a sliding spherical inhomogeneity, sub-
subjected to uniform tension at infinity.
Spheroidal inhomogeneous inclusions
Consider a spheroidal inhomogeneous inclusion Q in an infinite isotropic
body, where aY = a2 and the eigenstrain components in S2 are restricted to
?n = ?*2 ^ €зз so tnat tne symmetry about the хъ axis is conserved. The body
is subjected to all-around tension about the x3 axis or aniaxial tension in the
x3 direction in addition to the uniform eigenstrain e* in i2. The shear modulus
and Poisson's ratio are /x and v in the matrix and /I and v in п. The inclusion
can slide along the interface and no shear stress is allowed along the interface.
We follow the method developed by Edwards A951). Assume that п is
prolate as shown in fig. 51.1(a), where аг = a2 = a, a3 = b and b > a. When fl
is oblate (b< a), its solution can be predicted from the solution for the prolate
case according to Sadowsky and Sternberg A947).
The prolate spheroidal coordinate system (a, /3, y) is introduced as shown
in Fig. 51.1(b). The cartesian coordinates (x, y, z) are written as
x = с sinh a sin /3 cos у,
у = с sinh a sin /? sin у,
z = с cosh a cos /?,
E1.19)
where с is the focal distance in the z axis. The surface of the ellipsoid is
defined by a = a0.
Fig. 51.1. (a) A spheroidal inhomogeneous inclusion subjected to the applied load, (b) The prolate
spheroidal coordinate system.

51. Sliding inclusions 489
The displacement components in the cartesian coordinate system are ex-
expressed by the Boussinesg's potential functions \p, <p and X,
[u, v, w] = ^grad 4>,
[u, v, w] = -curl[0, 0, <p], E1.20)
The above formulae are for the matrix. The same formulae with bar-are used
for inertia points in fl. For this axial symmetry problem we can choose <p = 0.
The slip boundary conditions at the interface of the inclusion i.e. at a = a0
are
И-"Й0+--' Г'" E1-21)
where м* is the displacement caused by t* = e* and c* if the inclusion is free
from the surrounding medium.
The boundary conditions at infinity are
ox = oy=p0, a -» oo E1.22)
for the all-around tension case,
oz=px, a -> oo E1.23)
for the uniaxial tension along the z axis, and
°a = °fi = °y = 0 /
n E1-24)
та/3 = 0, a ^oo
for the eigenstrain problem, where the inclusion can be also an inhomogeneity.
Using Р„(р) and Qn{q), the Legendre functions of the first and second kinds,
we write
E1.25)

490 Chap. 7 Material properties and related topics
for the matrix (a> a0) and
n = 0
E1.26)
for the inclusion (a<a0), where
(p0 for all-around tension
рг for uniaxial tension
2це* or 2/xe* for eigenstrain case
and
q = cosh a, g = sinha,
p = cos /?, p = sin /?.
E1.27)
E1.28)
Fig 51.2 Variation of ол, ау and az (caused by all-around tension p0) in the inhomogeneity on
the spheroidal interface with <f> for s = 0.5, Г = 0 5, and у = \тт РВ stands for the perfect bonding
and SL for the sliding inclusion

51. Sliding inclusions
491
Fig 51 3. Variation of the stresses in Q (caused by tension pt in the z direction).
Fig. 51.4. Variation of ox, ay and oz (caused by eJ = e* = O, e* * 0) in the inclusion on the
spheroidal interface with <f> fors = 05, Г = 1 and y = \w

492 Chap. 7 Material properties and related topics
Numerical examples have been shown by Mura, Jasiuk and Tsuchida A985).
the solid curves in Fig. 51.2 ~ 4 are the stress components Q along the
interface and the dotted lines are the corresponding stress components for the
perfect bonding case, where Г = ju/ju and v = v = 0.3. It is interesting to
observe the deviation of the stresses from the uniformity.
52. Recent developments
Recent developments in inclusion problems can be reviewed from three points
of view:
1) applications to composite materials, including precipitates and martensite
problems in metallurgy,
2) a non-elastic matrix,
3) the fracture of composites and sliding and debonding.
Inclusions, precipitates, and composites
Average elastic moduli of composite materials, precipitates, and polycrystals
have been investigated using the same method.
Walpole A985) reviewed the problem of composite materials from the point
of view which emphasizes the early and enduring influence of the papers of
Eshelby A957,1959). Recent papers on the subject are either improvements of
approximations or slightly modified versions of approaches, e.g. Horii and
Nemat-Nasser A983,1985), Nunan and Keller A984), Nomura and Chou
A984), Aleksandrov and Olegin A985), Silovanyuk and Stadnik A984),
Mol'kov and Pebedrya A985), Kantor and Bergman A984), and Choi and
Earmme A986), among others.
For non-dilute concentrations of multiple constituents, the concept of an
effective (fictitious) third phase surrounding the constituents, which was
introduced by Christensen and Lo A979), has been developed by Benveniste
A985), Nomura and Oshima A985), Oshima and Nomura A985), and Hlavacek
A986), among others.
Mori-Tanaka's method A973) was written originally for the evaluation of
the average stress field in the matrix which contains dilute composites.
However, this method is also applicable to non-dilute concentrations of
constituents with slight modifications. Hatta and Taya A986) applied Eshelby's
equivalent inclusion method to evaluate the average heat conductivity of
composites with high concentrations of inhomogeneities. The average thermal
expansion coefficients of an aligned short-fiber composite were also evaluated
by Takao and Taya A985) using the same method. Weng A984) showed some

Inclusions, precipitates, and composites 493
correlation between results obtained by Mori-Tanaka's method and Hashin-
Shtrikman's method A963).
Mori-Tanaka's method A973) has been employed in a broad range of
inclusion problems, e.g. by Noyan A983) for equilibrium conditions for
average stresses measured by X-rays, by Stobbs and Paetke A985) for the
Bauschinger effect in cold drawn patented wire, by Sarosiek, Grujicic, and
owen A984) for the heterogeneity of deformation in the ferrite phase in steel,
and by Pedersen A983) for thermoelasticity and plasticity of composites,
among others.
Pore compressibility of materials containing voids, as well as overall elastic
moduli, is an important property. An analytical expression for the pore
compressibility of an isolated spheroidal cavity was given by Zimmerman
A985). Hashin A985) considered large isotropic deformations of composite
materials with porous media consisting of a finitely deforming elastic matrix
and spherical inclusions or voids; he showed that a very small number of pores
have a significant effect on expansion, but not on contraction.
Plastic behavior of polycrystals has been investigated by many researchers
after G.I. Taylor A938). The most recent work of Walpole A985) is on a
transversely isotropic aggregate of cubic crystals. For large plastic deforma-
deformations, Asaro and Needleman A985) presented a review on the subject.
Metallurgical consideration of precipitations in alloys and martensite trans-
transformations is also an important consequence of Eshelby's papers A957,1959).
The biasing precipitates of Ni4V were explained by the misfit of thermal
expansions and transformation strains by Wakashima, Ishige, and Umekawa
A982). The morphology of crystalline cubic precipitates in amorphous solids
was investigated by Schneck, Rokhlin, and Dariel A984) in terms of elastic
strain energy. The state and habit of Fe16N2 precipitates in b.c.c. iron were
studied by Hong, Wedge, and Morris A984). The habit plane for martensite
formation was investigated by Hayakawa and Oka A984), who evaluated the
minimum strain energy for various shapes and the lattice invariant shear.
The equivalent eigenstrains for inhomogeneities are misfit strains in a
homogeneous medium. The misfit strains have a tendency to decay when
diffusional relaxation takes place. Srolovitz, Petkovic-Luton, and Luton A983,
1984) and Srolovitz et al. A984) published a series of papers along this line.
Coated nuclear fuel particles and duplex cylindrical fibers in composite
materials are overlapping inhomogeneities. An overlapping inhomogeneity
consists of two confocal spheroids, and it is embedded in an infinite body.
When the coated region is thin compared with the size of inclusion, the
stresses in the overlapping inhomogeneity are approximated analytically as
done by Walpole A978). Such an assumption of the thickness has been
removed by Mikata and Taya A985,1986) for a spheroidal inhomogeneity and

494 Chap. 7 Material properties and related topics
by Theocaris, Sideridis, and Papanicolaou A985) for a cylindrical inhomogene-
ity. A four-concentric circular cylindrical model was used by Mikata and Taya
A985) for the stress analysis of coated fiber composites subjected to thermo-
mechanical loadings.
Half-spaces
Masumura and Chou A982) solved the antiplane eigenstrain problem of an
elliptic inclusion in an anisotropic half-space, and Zhang and Chou A985)
extended the analysis to the case of joint half-spaces with different anisotro-
pies. They obtained explicit expressions for the stress field and strain energy
under a given symmetry of the anisotropy of materials and the orientation of
the inclusion.
Selvadurai A985) developed a set of bounds which can be used to estimate
the asymmetric rotational stiffness of a rigid elliptical disc inclusion which is
embedded in bonded contact at an isotropic bi-material elastic interface.
The thermoelastic stress generated by surface parallelepiped inclusions has
been solved by Lee and Hsu A985) by the use of Mindlin's Green's function
for the half-space.
Loges, Michel, and Christ A985) have dealt with the chemical interaction of
a spheroidal inclusion with the surface of a half-space. They investigated the
dependency of energy on the shape, size, and orientation of the detect.
Non-elastic matrices
Hitherto, plastic deformations in matrices have been assumed to be uniform in
mathematical analysis when Eshelby's theory is applied to composites.
A numerical solution of local stress and strain fields near a spherical elastic
inclusion was obtained by Thomson and Hancock A984) when a matrix is
subjected to large, remote strains associated with the constitutive laws of
power hardening. Gilormini and Montheillet A986) calculated the stress
concentrations and deformations of an ellipsoidal inclusion when the matrix
and inhomogeneity have different viscous laws and applied stresses are sym-
symmetrical with respect to the axes of the ellipsoidal inclusion. When the matrix
is linearly viscoelastic, the stress field within the ellipsoidal inclusion is
uniform. Budiansky, Hutchinson, and Slutsky A982), Budiansky A983), and
Duva A986) have presented some basic studies of void growth and collapse in
non-elastic solids under biaxial loading at infinity. The inhomogeneity is a
spherical void and the matrix is a nonlinear, incompressible viscous material.
(The effective strain rate is proportional to the power of the effective stress.)
For polycrystalline metals undergoing creep at high temperatures, the
nucleation, growth, and coalescence of grain boundary cavities become im-

Cracks and inclusions 495
portant. Tvergaard A985) has shown a number of numerical solutions of
axisymmetric model problems to study the combined influence of sliding and
cavitation.
Grain boundary sliding versus intergranular plastic deformation requires
strain compatibility conditions. Mussot, Rey, and Zaoui A985) have shown a
geometrical analysis for the compatibility conditions.
Material damping of randomly oriented, short-fiber composites was
analyzed by Sun, Wu, and Gibson A985). Nomura and Chou A985) investi-
investigated the viscoelastic behaviour of short-fiber composite materials.
Dragon A985) described void growth around a spherical inclusion with the
use of the rate of a second-order damage tensor, whose definition supposes an
appropriate averaging process over a statistically significant, representative
volume. Descriptions of damage in materials by parameters in continuum
mechanical terms have been proposed by many researchers, e.g. Kachanov
A958), Krajcinovic A984), Jansson and Stigh A985), and Onat A984), among
others. The influence of matrix degradation on fiber-mode mechanical proper-
properties of unidirectional composites was investigated by Gottesman and Mikulin-
sky A984), who used the statistical parameters of the Weibull distribution of
the fiber strength.
Taya, Hall, and Yoon A985) gave experimental and analytical results of
void growth in a single crystal subjected to high strain-rate impact.
Delamination bucklmg of fiber-reinforced and laminated composites sub-
subjected to compressive loads along the reinforcing direction has been investi-
investigated by Budiansky A983) and Barber and Triantafyllidis A985).
Cracks and inclusions
Budiansky, Hutchinson, and Lambropoulos A983) introduced a continuum
description for an elastic solid which contains particles that undergo an
irreversible, stress-induced dilatent transformation. Stress and strain fields in
the transformation zone at the tip of a macroscopic crack are determined for
stationary and growing cracks. Detailed calculations for the toughening are
made over a wide range of possible material parameters.
Rose A986) and Rose and Swain A986) investigated enhanced fracture
toughness due to stress-induced transformation. Two viewpoints are proposed:
1) the increase can be attributed to the need to supply a work of transforma-
transformation, and 2) the transformation can be considered to result in internal stresses
which oppose the crack opening.
Budiansky, Hutchinson, and Evans A986) investigated a fiber-reinforced
ceramic subjected to tensile stress in the fiber direction. Extensive matrix
cracking normal to the fibers is considered under two distinct situations

496 Chap. 7 Material properties and related topics
concerning the fiber-matrix interface: 1) unbounded fibers initially held in a
matrix by thermal mismatches, but susceptible to frictional slip, and 2) fibers
that initially are weakly bonded to the matrix, but may be debonded by the
stresses near the tip of an advancing matrix crack.
A tensile crack bridged by fibers has been modeled in such a manner that it
can be analyzed based on concepts from the mechanics of inclusion. Mori and
Mura A984) have analytical expressions for the energy release rate, stress
intensity factor, and crack opening as functions of the size and spacing of the
fibers.
Hashm A985) employed the principle of minimum complimentary energy
to analyze cross-ply laminates which contain distributions of intralaminar
cracks within the 90 ° ply for tensile and shear membrane loading.
The energy release rates of various microcracks in short-fiber composites
have been discussed by Taya and Chou A983).
Characteristics of stress singularity at rigid inclusion tips have been in-
investigated by Wang, Zhang, and Chou A985) when the matrix is isotropically
elastic. Hasebe, Nemat-Nasser, and Keer A984) considered a kinked crack
initiating from a rigid-line inhomogeneity.
When a ribbon-like rigid inclusion is embedded in a power-law hardening
material, the order of the singularity at the ends of the inclusion under load is
exactly the same as that at the crack tips, according to Hayashi A982, 1984).
Elastic-plastic finite element analyses were used by Trantina and Barishpol-
sky A984) to compute the crack driving force for cracks initiating at voids and
inclusions.
Kunin and Gommerstadt A985) have given a general theory for the
interaction between a crack and an inhomogeneous inclusion by the use of the
projection integral equation method developed by Kunin A983).
Theocans and Demakos A985) considered an antiplane shear crack in an
infinite plate with a circular inhomogeneity. A finite element analysis of the
influence of an inhomogeneity or a hole on the stress intensity factor at the tip
of an edge crack in a three-point bending specimen was presented by Sides,
Perl, and Uzan A984). The interaction problem between a planar crack and a
flat inhomogeneity in an elastic solid was considered by Liu and Erdogan
A986). The stress intensity factors were calculated and tabulated for various
crack-inhomogeneity geometries, the inhomogeneity to matrix modulus ratios,
and general homogeneous loading conditions.
Expansion of a penny-shaped crack by an inclusion is a problem of
overlapping inhomogeneities. Selvadurai and Singh A984, 1986) evaluated the
stress intensity factor at a crack tip when the inclusion is a rigid circular disc.
When the inclusion is ellipsoidal, the contact area with the crack surface
becomes an unknown quantity. Tsai A984) solved the case when the inclusion
is an oblate, rigid spheroid and the matrix is transversely isotropic.

Dynamic cases 497
Sliding and debonding inclusions
The analysis for sliding inclusions has been extended to the nonshear eigen-
strain case by Mura, Jasiuk, and Tsuchida A985). They used Boussinesq's
potential functions and evaluated the displacement and stress fields when a
spheroidal inhomogeneity is subjected to all-around tension or uniaxial ten-
tension. Tsuchida, Mura and Dundurs A986) considered the two dimensional
elliptic inclusion with a slipping interface, and Leo et al. A985) evaluated
elastic fields about a spherical inclusion with a greased interface. Kouris,
Tsuchida, and Mura A986) showed that the result of a sliding circular
inclusion cannot be obtained as a limit of an elliptical sliding inclusion.
The problem of the stress field of an elliptic inhomogeneity which has
debonded over an arc along the interface has been considered by Karihaloo
and Viswanathan A985). They used Eshelby's equivalent inclusion method.
A more interesting problem was solved in 1971 by Hussain and Pu A971).
They considered an unbonded circular inhomogeneity in an infinite medium
subjected to uniaxial tension. The interface between the inclusion and the
matrix is partitioned into regions of rigid linkage, slip, and separation.
Solutions have been obtained by using variational method techniques devel-
developed by Noble and Hussain A969). Mura and Taya A985) introduced the
concept of Somigliana dislocations to describe plastic deformations on the
interface, and they evaluated residual stresses in metal composites due to
temperature change.
The effect of sliding on the overall behavior of composite materials has
been investigated by Benvemste A984) and Benveniste and Aboudi A984).
Second-phase particles or inclusions on a grain boundary suppress grain
boundary sliding. Mori et al. A983) modeled the grain boundary as a flat
ellipsoidal inclusion and the particles as disc-shaped inclusions which have
opposite eigenstrains from those given in the grain boundary, so that no
plastic deformation exist in the particles. Their theory has been proven in
experiments by Shigenaka, Monzen, and Mori A983) and Suzuki, Tanaka, and
Mori A985), and it has been extended to a crack arrest mechanism in
fiber-reinforced materials by Mori and Mura A984). A review on the subject
was given by Mori A985).
Dynamic cases
Disturbances of elastic waves by an embedded cavity have been investigated
by many authors. Bostram and Karlsson A985) evaluated point-force excita-
excitation of an elastic plate with an embedded cavity.

498 Chap. 7 Material properties and related topics
The equivalent inclusion method in the dynamic case was applied to the
scattering fields of two ellipsoidal inhomogeneities by Li, Zhong, and Li
A985).
When the inhomogeneity is rigid and the zeroth and first-order low
frequency approximations are used, the leading term of the scattering cross
section was obtained explicitly by Dassios and Kiriaki A986). Olsson A985)
used the null field approach for a star-shaped rigid inclusion.
Niwa, Hirose, and Kitahara A986) used the boundary integral equation
method for various inclusions in a half-space, such as a cavity, an elastic
inhomogeneity, and a fluid inclusion.
A new version of the effective field approach was presented by Beltzer and
Brauner A986) with application to random fibrous composites, where wave
velocities and attenuations are evaluated numerically. Elastic scattering prob-
problems have been investigated by Kinra and Li A986) for a random distribution
of inclusions and by Baik and Thompson A985) for a planar distribution of
inclusions. Coussy A986) considered the scattering of SH-waves by a rigid
elliptic fiber partially debonded from its surrounding matrix.
Miscellaneous
Eldiwany and Wheeler A986) studied the three-dimensional problem of
finding the shape of the minimum stress concentration for a rigid inhomogene-
inhomogeneity embedded in an infinite elastic matrix under uniform applied stresses at
infinity. The optimum inhomogeneities were found to be ellipsoidal in shape.
Castles and Mura A985) found that the displacement and stress fields
vanish in ?2 when eigenstrains are distributed in the form of 1/ |x | p outside of
?2, where ?2 is an ellipsoidal sub-domain in an infinite medium and | x | is the
distance from the center of ?2 and p > 0.
Dryden, Deakin, and Shinozaki A986) studied polymer spherulites which
have crossed polar and spherically symmetric elastic constants. They presented
a singular perturbation analysis of axisymmetric deformation.
When an inhomogeneity has the shape of an arc with a thin thickness, the
state of stress by Bernar and Opanasovich A984) is obtained as a solution of
two singular integral-differential equations of the Prandtle type.
The plane elasticity problem of an edge dislocation located within an
elliptical inhomogeneity was considered by Warren A983). Santare and Keer
A986) solved the case when an edge dislocation is located near a rigid
elliptical inhomogeneity. Contour plots for the glide component of the Peach-
Koehler forces were presented. Their solution has a closed form. Stagni and
I .izzio A984) obtained an infinite series form of the solution when an edge
dislocation is located near a general elliptic inhomogeneity.

499
Appendix 1
Einstein summation convention
If an index occurs twice in any one term, summation is taken from 1 to 3. For
instance,
Cijklekl ~ ?;,1/*1/ ¦*" 4,2/*2/ + ?|,3/*3/
= (?,711*11 + ?,712*12 + C,yl3e13)
+ Dy21e21 + Цу22е22 ¦*" 4/23^23 j
a/2,2
Kronecker delta
(A1 -
Namely, 5n = S22 = 833 = 1, 512 = 823 = 531 = 521 = 532 = Sl3 = 0. We have
«„ = 3. (A1.4)
Permutation tensor
A for the even permutation of 1, 2, 3,
-1 for the odd permutation of 1, 2, 3, (A1.5)
0 for other cases.

500 Appendix 1
Namely, €123 = t231 = e312 = 1, e132 = ?213 = e321 = -1,
?112 = ?223 ~ €333 =
We have

501
Appendix 2
The elastic moduli
The elastic moduli for isotropic materials are
Cijkl = ИД/ + MM,/ + tfifijk.
where Л and /i are Lame's constants. The following relations hold among
Young's modulus E, Poisson's ratio v, the bulk modulus K, and the shear
modulus ц:
A = 2/11-/A-2*), E=2(l + v)[i, p /( ^)
(A2.2)
^ = jB/t + ЗЛ) = ?/3A - 27), »- = CK-2ft)/2CK+ p).
Hooke's law becomes
+ \екк, (А2 3)
аи = 2це12,
о23 = 2це23,
or
а.-^гм^ + ^Ае^. (А2.4)
The inverse of (A2.3) are
en= {ап-"(о22 + а
«33=

502 Appendix 2
eu = аи/2ц, (А2.5)
When the reduced strain and stress are defined as
(А2.6)
Hooke's law can be written as
(A2.7)
The elastic moduli for cubic crystals are
ciJkl = ЧА/ + м*,Л + гЩк + м V/.
where all Sijkl = 0, except Snn = 52222 = 533зз = 1, and
X=CX2, y,= C^, ц' = Сп-Си-2С„. (А2.9)
These Ctj are the Voigt constants and are related to CtJkl as shown in Table
A2.1 for general cases. For instance, Cm2 = C12, C1123 = C14. Since Ciy = Cyj,
the number of independent constants is generally 21.
/ Relation between constant Cijkl and the Voigt elastic constants C:J
_
11 22 33 23 31 12
12 *-1Ч <-1d ^lS *--lft
11
22
33
23
31
12
C\\
C21
C3,
Qi
c51
Qi
C23
C33
C43
C53
c63
C24
C34
Cm
c54
Q4
C25
C35
c45
c55
c65
c26
c36
C46
c56
c66

Appendix 2 503
Let us make the notational changes,
«!! = «!, e12 = e2, e33 = e3, e23 = \e4, e31 = ?e5, el2 = \e6, (A2.10)
and
all=CTl> a22 = a2> °33 = <T3> CT23 = O4> CT31 = CT5> a12 = O6- (A2.ll)
Hooke's law is written as
0i = Cuex + C12<?2 + Cl3e3 + C14e4 + Cl5e5 + Cue6,
C22e2 + C23e3 + C24e4 + C25e5 + C26e6,
СЪ2е2 + С33е3 + C34e4 + C35e5 + C36e6,
C42e2 + C43e3 + C44e4 + C45e5 + C46e6, С • 2)
a5 = C51ej + C52e2 + C53e3 + C54e4 + C55e5 + C56e6,
a6 = C61e, + C62e2 + C63e3 + CMe4 + C65e5 + Q6e6.
For isotropic materials all C,7 = 0, except for
Сц = Q2 = Ц}з>
C12 = C13 = C23, (A2.13)
Q4 = Q5= Qe= 2(^11 ~ ^2)-
For cubic crystals (Al, Cu, Au, Fe, Pb, Ni, Ag, W, UO2, etc.) all CtJ = 0, ex-
except for
^¦11 = ^22 = ^33'
Q2 = C13=C23, (A2.14)
when the coordinate system is chosen along the crystalline directions. For
hexagonal crystals (Mg, Zn, Co, Cd, A12O3, etc.) all Ci} = 0, except for
^-11 ^ ^22 > ^33'
CU,C13 = C23, (A2.15)
O4= Qs > Qe= 2 (Qi ~ Q2) >
when the x3-direction is chosen along the hexad axis. This case is transversely
isotropic.

504 Appendix 2
For tetragonal crystals (Sn, Zr, MgCl, etc.) all C,y = 0, except for
Ql = Ог> Оз>
Cl2,Cu = C23, (A2.16)
Q4= Os > Oe >
when the x3-direction is chosen along the hexad axis. For orthorhombic
crystals (S, U, etc.), all CtJ = 0, except for
Ql> ^22> Оз>
C12, C13, C23, (A2.17)
Оч> Q5> Об>
when the coordinate system is chosen along the crystalline directions.
The numerical values of C,7 for various materials are found in Simmons
and Wang A971).
The elastic strain can be expressed in the stress by
e^s^ (A2.18)
or
eu = sijklokl,
where stJ is called the elastic compliance. CtJ is sometimes called the elastic
stiffness or the elastic constants.
There is a relation
*66 = 2Eп-*12) (А2.19)
when
Об = КС„-С12). (А2.20)

505
Appendix 3
Fourier series and integrals
Every function which is piecewise smooth in the interval —L<x<L and
periodic with the period 2L may be expanded in a Fourier series, that is,
f(x) = \ao+ ? [flncos—+ *>„ an —J, (A3.1)
where
j
1 -L
dx, (A3.2)
The convergence of the Fourier series is uniform in every closed interval in
which the function is continuous.
Equations (A3.1) and (A3.2) may also be written in the form
oo
/(*)= Г aBexp(ii!*x/L), (A3.3)
a"= JL /_V(x) exp( ~i

506 Appendix 3
To prove this, we derive (A3.1) from (A3.3). Substituting (A3.4) into (A3.3),
we have
OO-i
fix) = ? 2l/ f@ exp(-mvt/L) dt exp(inirx/L)
1 L
тт /" fit)cos(nmt/L)dt cos(nvx/L)
4т Г f
(A3.5)
since the terms cos,{nirt/L) sin(nirx/L) cancel the terms cos(-«77r/L)
sin(-/iwx/L). Furthermore, the identities cos(hu7/L) cos(«ttx/-L) =
cos(-и7T?/L) cos(-nwx/L), etc. give the equivalency between (A3.5) and
(A3.1).
It seems desirable to let L go to oo, since then it is no longer necessary to
require that / be continued periodically. We assume that f(x) is piecewise
smooth in every finite interval and that at the discontinuities, the value of the
function is the arithmetic mean of the right-hand and left-hand limits. The
further assumption that the integral ffx\f(x)\ dx exists is added. Setting
m/L = 8, (A3.3) and (A3.4) become
OO
/(*)= I а„ехр(ш8х), (А3.6)
dx. (A3.7)
Furthermore, we set nS = ?, S = d? and let L -» oo and 5 -> 0. Then, (A3.6)
can be written as
fix) = Г f(t) exp(/?x) d|, (A3.8)
J — 00
where

Appendix 3 507
The expression (A3.8) with (A3.9) is called the Fourier integral form of f(x).
/(?) is called the Fourier transform of f(x).
Dirac's delta function and Heaviside's unit function
Dirac's delta function 8{x) has the property
Г 8{x)F(x) dx = F@) (A3.10)
J — oo
for any suitably continuous function F(x). 8(x) may be defined by
sin nx , . /АЭ11Ч
hm = o(x). (A3.ll)
n —* oo fTX
The Fourier transform of 8(x) is obtained from (A3.9) as
8@ = A/2»). (A3.12)
The Fourier integral form of 8(x) is, therefore,
8(x) = (l/2ir) Г exp(ifce)d?. (A3.13)
*-00
Although the integral in (A3.13) does not converge, the formality (A3.13)
holds. 8(x) is the generalized function (see Lighthill 1964).
Heaviside's unit (or step) function H(x) is defined by
The Fourier transform of H(x) becomes, from (A3.9),
Я@ = l/2m? (A3.15)
if one can justify throwing out the term exp( — i?x) at |= oo. According to
Lighthill (p. 33, 1964), its correct expression is
where sgn ^ is defined as
1 for{>0,
-1 for КО.
(А3-17)

508
Laplace transform
Combining (A3.8) and (A3.9), we have
where t is used for x.
If
/@ = 0 forf<0,
= e-ytF(t) forf>0,
where у > 0, (A3.18) becomes
! = ¦=— / e'(t dU F(T)e'iy+i$)r d^
2nJ-<x> Jo
= -—:f ezt dz I F{r)e~ZT dT,
where z = 7 + /? is used.
The Laplace transform of a function F(f) is defined as
F(j)= f°°F(t)e-s'dt, s>0.
Appendix 3
(A3.18)
(A3.19)
(A3.20)
(A3.21)
When s is formally extended to a complex variable z, (A3.20) can be written
as
(A3.22)
which is called the inversion integral.
If F(s) is obtained as an analytic function in the semi-infinite plane
R(z) < у except at poles zn(n = 1,2,...) and
(-/3N<x<y, \y\<PN)
(A3.23)

Appendix 3 509
where SN -» 0, fiN -» oo for iV -» oo, (A3.22) becomes
*•(')= ?р„('), (А3.24)
where pn is the residue of ez'F(z) at 2И. All poles of ez'F(z) are located in the
plane R(z)< y.

510
Appendix 4
Dislocation pile-ups
The Hilbert integral equation for an unknown f(x),
Dx — i
has been investigated by Muskhelishvili A953) and illustrated for the disloca-
dislocation problems by Head and Louat A955). In the above equation a(x) is a
given piecewise function. It is understood that the Cauchy principal value of
the integral (see Whittaker and Watson A962, p. 75) is to be taken to avoid
divergence at t = x.
Let us assume that D consists of a finite number of intervals of the x-axis
(аи Ьг), (a2, b2),...,(ap, bp). Suppose that at q of the end-points of the
segments, denoted by cb c2,...,cq, /(x) is to remain bounded, and that at the
remaining Ip-q end-points, denoted by cq+l, cq+2,...,c2p, f(x) may be
unbounded. Let
(A4.2)
R2(x)= П (x-ck),
where Rt = 1 when q = 0, and R2 = 1 when 2p < q + 1. Then, if p — q > 0,
solutions of (A4.1), bounded at cu...,cq, always exist and are given by
(A4.3)

Appendix 4 511
where Qp-q-\(x) is an arbitrary polynomial of degree not greater than
p — q — 1. It is identically zero for p = q and p = q + 1.
If /? - q < 0, a unique solution, bounded at cx, . . . , cq exists if and only if
a(x) satisfies the conditions
) '"°(Od/ = o, « = o,i,-..,(q-p-V, (A4.4)
j
and if this is so, the solution is given by (A4.3) with Qp_q_1 = 0. Moreover at
a bounded end-point, f(x) vanishes. The Cauchy principal Values of the
integrals are taken in all these formulae.
Let us consider an example, a = a0 = constant, D = ( — с, с), and /(x) is
unbounded at — с and с Then, ax = —с, Ъх = с, р = 1, q = 0, cq+ г = —с, and
cq+2 = с We have R^ = 1, and R2(x) = (x + c)(x - c). Since p-q>0, (А4.3)
becomes
1
Qo is an arbitrary constant.
Another example will be considered: D = ( — a, a), a = a0 in |jc| <c,
a = a0 - к in с < | x \ < a, and f(x) is bounded at -a and a, where a0 and к
are given constants. Then, ax = -a, bx = a, p = 1, q = 2, q = -a, and c2 = a,
therefore R^x) = (x + a)(x - a), R2 = 1. Since p - q < 0,
a°dt
¦'с
^2
(а2-/2I/2(х-0
(A4.6)

512 Appendix 4
The condition (A4.4) becomes
Г d/ = 0. (A4.7)
J-a{a2-t2I/2
Further calculations for these integrals lead to
h- тех a — с \ A- r\ n — x \
"¦ x(a2 —c2) —c(a2 —.
кл x(a2 - c2I/2 + c(a2 - x2)l/1 c , , ... 0,
= — log— '-— '—- for x\<c (A4.8)
t2 c(a2 - x2I/2 - x(a2 - c2I/2
and
с/а = со${что°/2к). (А4.9)
The last result (A4.9) has been obtained from (A4.7).

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H. Zorski, Theory of discrete defects, Arch. Mech. Stos., 18 A966) 301-372.

Author index
Aaronson, H.I., 110, 219, 222
Abe, H., 307
Aboudi, J., 497
Achenbach, J.D., 319, 320, 33, 438, 472
Acton, J.R., 433
Adachi, H., 484
Adams, D.F., 433
Aderogba, К, 127
Airapetyan, V.M, 168
Aleksandrov, A.Y., 492
Alexander, D., 497
Aman, S, 348
Amehnckx, S, 44
American Institute of Physics Handbook,
252, 279
Anderson, D.L., 438, 455
Andreykiv, A.E., 298, 300, 307
Ang, D.D., 360
Anthony, K.H., 53
Ardell, A.J, 168
Argon, A.S., 110, 420, 447
Armstrong, R.W., 52, 356
Asaro, R.J., 48, 49, 92, 160, 169, 234, 235,
236, 270, 271, 273, 274, 290, 333, 334,
335, 337, 356, 412, 449, 485, 493
Ashby, M.F , 398, 420, 447
Ati, A., 48
Atkinson, C, 291, 322
Atkinson, J.D , 323, 402, 408
В
Backofen, W.A, 386
Bacon, D.J., 13, 49, 324, 335, 398
Baik, J, 498
Baker, В., 322
Baker, G S , 443
Banerjee, P.K., 298
Barber, J., 495
Barenblatt, G.I., 281
Banshpolsky, H., 496
Barnett, D.M, 13, 32, 49, 92, 160, 219,
222, 235, 270, 271, 273, 274, 290, 324,
333, 334, 335, 337, 354, 410, 412, 465,
469, 471, 493
Bassani, J.L., 447
Bechtoldt, C.J., 50
Beevers, R В., 49
Beltzer, A I, 378, 498
Benveniste, Y , 492, 497
Bergman, D.J., 492
Bernar, 11., 498
Berry, B.S, 168
Bhargava, R.D., 141
Bilby, В А, 7, 34, 65, 73, 229, 271, 288,
290, 314, 324, 335, 342, 343, 358, 364,
374, 384, 453
Bishop, J.F.W., 421, 442, 447
Boas, W., 421
Bodner, S.R, 364
Bonnet, R, 48
Born, M., 477
Bornett, A., 168
Bose, S K., 178, 459
Bostrom, A., 497
Bouhgand, Y., 50
Boussinesq, J., 177
Bozkurt, R.0, 123
Brauner, N, 498
Brebbia, G.A., 298
572

Author index
573
Bnerley, P., 453
Broberg, K.B., 322
Bross, H., 26, 348
Brown, L.M, 49, 169, 335, 336, 337, 338,
379, 392, 396, 401, 402, 408
Bruggeman, D A.G., 421
Budiansky, B, 313, 314, 431, 433, 438,
439, 445, 446, 453, 455, 494, 495
Bui, H D , 298, 307, 446
Bullough, R., 34, 73, 198, 324, 335, 342,
358, 465
Burgers, J.M., 13,17
Burgers, W.G., 234
Butterfield, R., 298
Byskov, E., 275
Clements, D L. 472
Clifton, R.J, 359, 360
Coffin, L F , 386
Cohen, J В , 21
Cole, D , 438, 455
Collins, W.D, 293
Colonnetti, G , 208, 212
Comninou, M, 338, 465, 472, 473, 474,
475, 476
Cook, R H , 169
Coussy, O., 498
Cottrell, A H., 288, 290, 324, 467
Cowin, S.C, 439
Craggs, J W., 322
Cruse, T.A., 298, 307
D
Calhas, C, 358
Campbell, G.A., 17
Castles, R.R, 498
Cesaro, E , 7, 65, 69
Charms, С С, 437, 438
Chan, S K., 275
Chang, S J, 291
Charsley, P, 387
Chen, C.H., 433
Chen, I.W, 447
Chen, S.H., 472, 475, 476
Chen, W.T, 141, 178, 270, 279
Cheng, D.H, 117
Cheng, PC, 151, 155, 168, 220, 223, 224,
251, 252, 277
Cheng, S., 433
Cherepanov, G P, 34
Chiu, Y P , 104, 105, 107, 121
Choi, В I., 492
Chou, T W, 29, 31, 50, 53, 338, 448, 492,
495, 496
Chou, Y T, 114, 128, 141, 142, 466, 467,
469, 494, 496
Chray, T.P., 237
Christ, A., 494
Chnstensen, R.M , 448, 492
Christian, J.W., 228, 229, 233
Chuang, T.J., 420
Cladis, P E., 49
Clarke, D.R., 401
Danel, M.P, 493
Das, E.S P., 53, 356
Das, S C, 178
Dassios, G, 498
Datta, S.K., 458, 459
Deakin, A S , 498
Demakos, C.B, 496
Desvaux, M P.E , 387
DeWit, R, 13, 52, 53, 324, 356
Dietze, H D , 368
Dixson, J.P, 375
Domany, E, 458, 459
Doner, D R., 433
Dragon, A., 495
Drucker, D C, 439
Dryden, J R , 498
Dugdale, D S, 281, 285
Dundurs, J., 43, 44, 53, 126,127, 338, 370,
463, 464, 465, 471, 472, 497
Duquette, D J., 379
Duva, J.M. 494
Dyson, FW., 85, 91, 95, 174
Earmme, Y Y , 198, 484, 492
Easterling, К Е , 231, 234
Eberhart, M E., 484
Ebner, M L., 386
Edelen, D.G.B., 353

574
Author index
Edil, Т., 439
Edwards, R H., 178, 488
Einspruch, N.G., 458
Eisenberg, M.A , 364, 367
Eldiwany, B.H, 498
Elhott, H.A., 29
Engel, J.J, 379
England, A.H., 293, 471, 475
Erdogan, F., 275, 319, 471, 474, 496
Enngen, A.C., 439
Eshelby, J D , 1, 7, 13, 26, 34, 36, 56, 74,
92, 100, 121, 141, 157, 168, 177, 206,
261, 271, 288, 311, 314, 322, 335, 343,
357, 358, 359, 360, 394, 449, 453, 466,
485, 492, 493
Essmann, U , 49, 53
Eubanks, R.A, 177
Eunn, P.H., 168
Evans, A.G, 495
Ezaki, H., 484
Faivre, G., 104
Feldman, E.P., 361
Ferrers, N M , 85, 95
Feshbach, H., 458
Field, F.A., 291
Fine, M.E., 317
Flinn, J.E., 378
Foreman, A.J E., 26
Foster, R M , 17
Frank, F C, 49, 57, 357, 466
Fredholm, I., 13, 26
Frenkel, Y.I., 358
Freund, L.B , 319, 320, 322
Friedel, J., 49, 324
Fu, W.S., 242, 460
Fung, Y С , 451
Furuhashi, R., 169, 170, 198, 486, 487
Futagami, T 298
Gale, В., 314
Galerkin, В., 177
Gangadharan, A.C., 464
Gardner, L.R.T., 364, 374
Gavazza, S D., 335, 356, 465
Gdoutos, E E., 286
Gerstner, R.W., 44
Geyer, J.F., 276
Ghahremani, F., 447, 488
Gibson, R.F., 495
Gilman, J.J , 50, 364, 387
Gilormini, P., 494
Glicksman, M.E., 465
Golebiewska-Lasota, A.A, 353, 356, 361
Gommerstadt, В., 496
Goncharyuk, I.V., 302
Goodier, J.N., 40, 74, 88, 170, 177, 291,
369
Gottesman, Т., 495
Gough, H.J, 177
Gould, D., 402
Gradshteyn, I S., 78
Granato, A, 361
Green, A.E., 141, 295
Griffith, A A., 241, 244
Grosskreutz, J.C., 379
Grujicic, M, 493
Gubernatis, J.E., 458, 459
Guell, D.L., 127
Giinther, H., 53, 348, 353, 359, 361
Gupta, G., 474
H
Hahn, G.T , 291, 323
Hahn, H.T., 364
Hall, I.W, 495
Ham, R.K., 169
Hancock, J.W., 494
Hardiman, J., 41
Harris, W F., 49, 50
Hart, E.W, 361, 398
Hasebe, N., 465, 496
Hashin, Z., 433, 434, 437, 448, 453, 493,
496
Hatta, H., 492
Havner, K.S., 443
Hayakawa, M., 493
Hayashi, K., 291, 307, 472, 496
Hayns, M R , 465
Hazzledine, P.M, 416
Head, A.K., 289, 466, 467, 510

Author index
575
Heald, P.T, 314
Heinisch, H.L., 465
Herrmann, G., 438
Hershey, A.V., 433, 439
Hetenyi, M., 126
Hlavacek, H, 492
Hill, R, 39, 42, 366, 421, 425, 426, 433,
436, 439, 442, 443, 445, 447, 449
Hilliard, J.E, 223
Hirose, S., 498
Hirose, Y, 387
Hirsch, P.B , 44, 402, 411
Hirth, JP, 13, 44, 49, 324, 333, 334, 335,
356
Hobson, E.W., 31, 162
Hodge, P G.P., 373
Hoemg, A, 265
Hollander, E.F , 348, 353, 361
Hong, H , 493
Honjo, G., 379
Horgan, CO, 438
Нош, Н., 276, 439, 492
Home, M.R, 439
Hoshina, M., 386
Howard, I C, 453
Howie, A., 44
Hsu, С С , 121, 494
Huang, К., 477
Huang, W., 53
Huber, A., 421
Huberman, M., 458
Hull, D., 324
Humphreys, F.J, 402
Hussain, M A., 497
Hutchmson, J W., 314, 323, 439, 443, 445,
446, 447, 449, 453, 494, 495
I
Ibaraki, M, 233
Ichikawa, M , 50, 310
Ikeda, H., 460
Im, J., 420
Indenbom, V L., 13, 26, 48, 335
Inglebert, G, 379
Inglis, С Е, 279, 293
Ioakimidis, N 1, 41
Irwin, G.R, 260, 274, 286, 309, 319
Ishige, К, 493
Ishioka, S, 360
Isida, M, 275
Ito, Y M , 379, 446
Iwakuma, Т., 443, 449
Iwan, J, 497
Izurm, Y, 317
Jack, K..H, 237
Jan, R.V, 26
Jansson, S., 495
Jasiuk, 1, 492, 497
Jaswon, M A , 141
Jaunzemis, W., 364
Jimma, T, 378
Johnson, К Н., 484
Johnson, W.C, 104, 108, 109, 198, 226,
227
Kachanov, L M , 495
Kagawa, К 1, 420
Kageyama, К , 291, 313
Kamei, A , 275, 310
Kamitani, A, 379
Kanmnen, M.F, 291, 319, 320, 323
Kantor, Y., 492
Kanhaloo, В L., 290, 291, 364, 497
Karlsson, A., 497
Kassir, M К, 244, 260, 265, 268, 270
Kato, M, 146, 168, 220, 223, 230, 235,
387
Kawai, К, 378
Kay, T R , 291
Keer, L M, 242, 276, 293, 297, 298, 465,
471, 472, 475, 476, 496, 498
Keith, R E, 387
Keller, J В , 492
Keller, J M., 338
Kelly, J M , 364
Lord Kelvin, 23
Kfoun, AP, 310, 311
Khachaturyan, A G , 9, 168, 233
KJietan, R P., 472
Kikuchi, R, 104
Kjm, T J , 41

576
Author index
Kinoshita, N, 26, 32, 135, 160, 169, 170,
298, 412
Kinra, V К , 498
Kinsman, К R., 110, 235, 236
Kirchner, H.O К , 356
Kiriaki, K., 498
Kitagawa, K.411
Kitahara, M., 298, 460, 498
Kiusalaas, J , 359, 360
Kleman, M , 49, 50
Klesnil, M., 386
Knauss, W.G , 309, 323
Kneer, G , 139, 140, 433
Knopoff, L, 458
Knowles, J К 313
Kobayashi, A S , 270
Kobayashi, H., 318
Kobayashi, S., 291, 460
Kochs, U.F., 398
Koda, M , 420, 497
Koehler, J S , 19, 26, 313, 355, 361
Koeller, R С , 419
Kohn, W., 438
Koiter, W Т., 274
Kondo, K., 7, 65, 73, 324
Kontorova, T, 357
Kornnga, J., 438, 439
Kosevich, A.M., 13, 348, 361
Kossecka, E, 52, 348
Kouns, D.A, 497
Krajcinovic, D, 495
Krishna Murty, A V., 274
Knzek, R.J., 439
Kroner, E, 1, 7, 13, 26, 29, 31, 32, 45, 53,
65, 73, 226, 227, 324, 340, 353, 361,
364, 431, 433, 439, 445, 446
Kroupa, F., 297, 338, 339
Krumhansl, J.A., 438, 458, 459
Kuang, J.G , 467, 471
Kundu, A.K., 453
Kunin, I.A, 13, 324, 484, 496
Kuo, H.H , 53, 452
Kupradze, V.C., 298, 460
Kunhara, T, 465
Kunyama, M., 477
Kusumoto, S., 275
Kuwata, Т., 464
Lachenbruch, A.H, 274
Lacht, J.C, 298
Laird, C, 104, 110, 318, 379
Lambermont, J H , 364
Lambropolous, J.C, 495
Lame, M G , 177
Langhaar, H.L., 42
Lardner, R W , 314, 324, 348
Laszlo, F, 220
Latanision, R.M., 484
Laub, T, 359
Laws, N., 155, 433, 453
Lee, E.H., 438
Lee, J D , 309
Lee, J.K., 104, 108, 109, 198, 214, 222,
226, 227
Lee, M S., 358, 464, 471
Lee, N G., 114
Lee, S , 121, 494
Leibfned, G., 13, 358, 359, 467
LejSek, L, 29
Lekhnitskii, S.G., 279
Leo, P-H., 497
Leon, A, 177
Leslie, W C, 237
Levy, M, 363
Lewis, J L , 475
Li, G F., 498
Li, H., 498
Li, J С М , 50, 53, 291, 466, 467
Li, P, 498
Lie, K.H.C., 26
Lieberman, D.S., 230, 232
Liebowitz, H., 260, 309
Lifshitz, I M , 26, 29
Lighthill, M.J., 13, 507
Lin, СТ., 314
Lin, L.H., 292
Lin, SC, 137, 217, 222, 223, 251, 252,
401, 465
Lin, T H., 121, 379, 439, 445, 446
Ling, C.B, 177
List, R.D, 141
Liu, G.C.T, 53
Liu, X.H., 496
Lizzio, R., 498

Author index
577
Loges, F, 494
Lo, К Н , 492
Lothe, J, 13, 44, 49, 324, 333, 334, 335,
336, 338, 359, 361
Love, A.E H., 177
Lowengrub, M , 260, 476
Lu, T.L, 338
Lucke, К, 361
Luk, C.H., 275
Lukas, P, 286
Luque, R F., 439
Lune, A T , 178
Luton, M J , 493
M
MacMillan, W.D., 108
Мак, А.К., 475
Mai, A.K , 458, 459
Malen, К , 335, 348
Mandel, J., 439, 446
Mandl, G., 439
Mann, E, 20
Manson, S.S, 386
Maradudin, A.A., 477, 482
Marcoh, G, 48
Markenscoff, X , 359, 360
Martin, D E , 386
Martynenko, M.D, 298
Marukawa, K., 53
Marcinkowski, M J , 53, 356
Mason, W P., 360
Mastrojannis, E N., 299
Masuda, T, 378
Masumura, R.A , 465, 494
Matsuoka, A., 464
McCartney, L N.. 314
McClintock, F.A., 311, 420
McCoy, J.J., 459
McLaughhn, R , 433, 453
Mehrabadi, M.M., 439
Meshii, M, 218
Michel, В., 494
Michell, J.H., 43
Miekk-oja, H.M, 234
Mikata, Y., 42, 493, 494
Mikhhn, S.G., 298
Mikulinsky, M., 495
Milne-Thomson, L.M., 41
Minagawa, S, 53, 348, 353, 356, 361, 438
Mindhn, R.D., 112, 113, 117, 177, 439,
494
Minster, В., 438, 455
Mirandy, L., 279
Mises, R von, 363
Miyamoto, H, 178, 187, 270, 279, 291,
313
Miyata, H., 275
Miyoshi, T, 270
Mochizuki, Т., 413
Mogami,T, 439
Montheillet, F., 494
Monzen, R, 411, 420, 497
Moon, F С , 128, 155
Mori, Т., 146, 168, 217, 218, 220, 223,
224, 230, 235, 238, 239, 328, 387, 392,
396, 401, 402, 405, 406, 407, 408, 411,
412, 413, 414, 419, 420, 492, 493, 496,
497
Monguti, S , 7, 65
Monnaga, H., 484
Monta, H., 291
Morkov, V.A., 492
Morns, J.W., 493
Morns, P R., 433
Morse, P.M., 458
Moschovidis, Z A., 39, 160, 191, 196
Mossakovskii, V.I., 476
Mott, N F., 379
Mow, C.C., 458
Mroz, Z., 439
Mughrabi, H., 379
Mikherjee, A.K., 291, 323
Мига, Т., 9, 11, 21, 26, 32, 47, 48, 51, 53,
78, 117, 121, 122, 123, 135, 137, 146,
151, 155, 160, 168, 169, 170, 178, 187,
198, 217, 217, 220, 222, 223, 224, 230,
235, 251, 252, 270, 277, 286, 293, 297,
298, 313, 314, 318, 324, 328, 338, 343,
345, 348, 350, 352, 356, 359, 360, 364,
367, 370, 371, 376, 378, 379, 382, 387,
401, 406, 411, 412, 419, 420, 438, 439,
447, 452, 459, 460, 463, 465, 469, 471,
484, 486, 487, 492, 496, 497, 498
Murakami, Y., 270, 275
Muskhelishvili, N 1, 41, 187, 510
Mussot, P, 495

578
Author index
N
Nabarro, FRN, 44, 49, 226, 324, 359,
360, 466
Nagakura, S., 226
Nakada, Y , 237
Nakahara, 1, 178
Nakai, Y., 314
Narayan, J, 291
Nanta, K., 402, 408
Needleman, A., 449, 493
Nemat-Nasser, S., 276, 291, 438, 439, 443,
449, 465, 472, 492, 496
Nemenyi, P , 7, 45
Neuber, H, 177, 187, 279
Neuman, P., 318, 379
Nicholson, R В , 168
Niesel, W, 178
Ninomiya, T , 359, 484
Nishimura, N , 460
Nisitani, H , 270, 274
Niwa, Y., 460, 498
Nix, W D , 493
Noble, В, 497
Nomura, S , 492, 495
Novakovic, A, 218
Nowick, A S , 168
Noyan, I C , 493
Nuismer, R J , 320
Numan, K.C, 492
Nye, J.F, 53, 338, 364, 373
О
O'Connell, R J , 438, 455
Oda, M , 439
Ohnami, M , 379
Ohr, S M , 291, 465
Oka, H , 493
Okabe, M , 411, 413, 414, 419
Okamura, H , 291
Olegin, I R , 492
Olsson, P, 498
Onat, ET.495
Onaka, S., 387, 411
Ono, К, 102, 231, 233, 234
Opanasovich, V K. , 498
Oranratnachai, A, 476
Orlov, S S , 26, 48, 335
Orowan, F , 44, 286, 342, 361
Orsay Liquid Crystal Group, 49
Osawa, T, 412, 413, 414
Oshima, N , 439, 492
Owen, D R J , 122, 123, 338, 379
Owen, W S , 493
Paetke, S, 493
Palaruswamy, К, 309
Pan, Y.C, 29, 31, 114, 338
Panasyuk, V V , 298, 300
Pao, Y H , 128, 155, 458
Papanicolaou, G С , 494
Papkovich, PF, 177
Panhar, К S , 276
Paris, P.C , 260, 314, 317
Parnes, R., 378
Pastur, L A., 361
Paul, В, 279
Peach, M.0, 313, 355
Pebedrya, В Е, 492
Pedersen, О В , 493
Pegel, В , 359
Peierls, R.E, 358
Penisson, J M , 168
Perl, M, 495
Peterson, R E , 187
Petkovic-Luton, R A., 493
Phillips, A., 364, 367, 446
Pian, T H H , 275
Pieranski, P, 49
Polak, J, 386
Polanyi, M., 44
Pook, L P , 275
Prager, W., 373, 439
Prandtl, L, 363
Pu, S.L., 497
Purdy, G R., 169
R
Radhaknshna, HC, 141
Raj, R., 419, 420, 447
Raju, I.S, 275
Rao, А К , 275

Author index
579
Read, T A , 230, 232
Read, W.T, 19, 26, 34, 36, 141, 273, 324,
335, 374
Reissner, H, 1, 7, 65
Reuss, A., 363, 421, 424
Rey, C, 495
Rice, J.R, 310, 311, 313, 314, 323, 420,
443, 449, 465, 471
Riedel, H , 291
Rieder, G , 73
Rizzo, F.J, 298
Robinson, С, 49
Robinson, K., 178
Rokhlin, S 1, 493
Rongved, L, 124
Rose, L.R F , 495
Rosenfield, A R , 291, 323
Rosengren, G F , 314, 323
Routh, EJ., 77
Rozenzweig, L N., 26, 29
Rudmcki, J.W , 448, 449
Russell, K.C, 219
Rvachev, V L., 302, 303
Rykba, M T , 476
Ryzhik, I M , 78
Saada, G , 338, 361
Sack, R A , 242
Sadowsky, M A, 177, 178, 196, 267, 279,
488
Safoglu, R, 420
Saito, K., 123, 338, 348
Saito, Y., 178
Salamon, N J., 338
Sankaran, R., 104, 110
Santare, M.H., 498
Sarosiek, A M, 493
Sass, S.L, 21
Satake, M, 439
Sato, A., 238, 239, 411, 420
Savin, G N , 187
Scattergood, RO, 13, 49, 324, 335, 398
Schaefer, H , 53, 353, 361
Schmid, E., 421
Schmueser, D., 472
Schneck, R, 493
Scriven, L E , 50
Seeger, A , 26, 53, 324
Sekerka, R F , 497
Seidel, E.D, 453
Sekine, H, 270, 338, 465
Selvadurai, A P.S , 178, 494, 496
Sendeckyj, GP, 41, 89, 433, 437, 438,
463, 465
Seo, К, 117
Shah, R С, 270
Shalaby, A H , 443
Shapiro, G S , 177
Sherman, D.I, 41
Shetty, D К , 218
Shibata, M, 102, 217, 231, 233, 234, 401
Shield, RT, 265,439
Shigenaka, N , 497
Shinozaki, D M, 498
Shintani, К, 361
Shioin, J., 378
Shioya, S, 464
Shioya, Т., 378
Shiozawa, К, 379
Shockley, W., 26, 34, 36, 141, 335
Shokooh, A , 276, 439
Shtnkman, S., 433, 434, 493
Sidendis, E P., 494
Sides, A, 496
Sih, G С, 244, 260, 265, 268, 270, 472
Silovanyuk, V P , 492
Sills, L В , 420
Simmons, G, 504
Simmons, J.A., 324
Sines, G., 104, 465
Singh, B.M, 496
Slutsky, S, 453, 494
Smith, С S , 378
Smith, E., 73, 246, 290, 364, 374, 469
Smith, G С, 318
Smith, J, 50
Sneddon, IN , 242, 260, 267, 299, 476
Sobezyk, К, 459
Sokolmkoff, I S , 72
Sokolovsky, V V., 371
Sormgliana, С, 13, 261, 270
Sorensen, E P., 311
Southwell, R V , 177
Spencer, A J M , 439

580
Author index
Sprys, J W , 234, 235, 236
Srolovitz, D J., 493
Stadnik, M M., 298, 300, 307, 492
Stagiu, 1, 498
Steeds, JW, 13, 324, 335
Steketee, J.A, 13
Steinberg, E, 177,178, 196, 267, 279, 313,
488
Stigh, U , 495
Stipps, M , 465
Stobbs, W.M , 392, 401, 402, 408, 493
Stokes, G.G, 64
Storen, S., 449
Stroh, A.N., 34, 37, 335, 467
Suezawa, M, 233
Sum, Y , 275, 276
Sun, С Т., 438, 495
Suzuki, H, 324
Suzuki, K., 420, 497
Sveklo, V A , 31
Swain, M V , 495
Swanger, L.A , 271, 335
Swinden, К H , 288, 290
Synge, J.L, 26
Tada, H., 275
Tagaya, M, 233
Taira, S, 386
Takao, Y , 448, 492
Takeuchi, T, 291
Tamate, О , 465, 471
Tamura, 1, 233
Tanaka, Ke, 88, 187, 314, 379, 382
Tanaka, Ко., 218, 392, 396, 401, 492, 493
Tanaka, M, 298
Tanaka, R, 497
Tanaka, Y., 238, 239
Taya, M , 42, 48, 447, 448, 453, 492, 493,
494, 495, 496, 497
Taylor, G 1, 19, 44, 439, 442, 493
Teodosiu, C, 13, 47
Tetelman, A.S , 469, 471
Teutonico, L J., 358
Theocaris, P S , 41, 286, 494, 496
Tholen, A R., 231
Thomas, S.L, 50
Thomas, T.Y., 449
Thompson, A W , 386
Thompson, D D., 439
Thomson, P.F., 467
Thomson, R., 291, 292, 465
Thomson, R D., 494
Timoshenko, S , 88, 170, 369
Tokushige, H., 401, 405, 407, 408, 411
Tomkins, В., 386
Tong, P., 275
Toya, M, 471
Toyoshima, M, 226
Trantina, G G., 496
Trauble, H , 49, 53
TnantafylUdis, N., 495
Truell, R, 458
Tsai, Y.M., 496
Tsuchida, E, 121, 178, 492, 497
Tuba, I.S, 275
Tucker, M O., 471
Tung, T К , 121
Tvergaad, V., 495
U
Ukadgaonker, V G, 41
Umekawa, S, 493
Uzan, J., 496
Varadan, V К , 459
Varadan, V V , 459
Varatharajulu, V, 459
Vause, R F., 443
Vilmann, C, 286, 371
Visscher, W M., 459
Viswanathan, К, 497
Vitek, V., 291
Voigt, W , 421
Volterra, V., 7, 13, 44, 46, 49, 65
W
Wakashima, K, 401, 493
Walpole, L.J , 39, 42, 433, 492, 493
Walsh, J.B., 438, 455
Wang, H., 504

Author index
581
Wang, Z.Y, 496
Ward, J.C., 49
Warren, W.E, 498
Waterman, P.C, 459
Watson, G.N., 164, 170, 510
Watson, J O., 298
Watwood, V B. 275
Wayman, СМ., 229, 231
Weaver, J., 298, 307
Wechsler, M.S , 230, 232
Wedge, D.E., 493
Weeks, R., 465
Weertman, J, 286, 291, 292, 314, 318,
324, 358, 359
Weertman, J R., 324
Weiner, J.H., 360, 484
Weingarten, G , 7, 13, 44, 65
Weng, G J , 364, 367, 446, 447, 492
Werne, R.W., 364
Westergaard, L.T., 274
Wheeler, L Т., 498
Wheeler, P, 438
Whelan, M.J., 41, 44
Whittaker, ET., 164, 170, 510
Wigglesworth, L.A., 274
Williams, С Е., 49
Williams, S.C, 233
Williams, MX, 275, 323, 360, 471
Willis, J R., 18, 26, 32, 47, 55, 141, 198,
260, 262, 265, 270, 271, 325, 328, 333,
433, 435, 460, 465, 476
Wilson, W.K, 275
Witterholt, E.J, 458
Wnuk, M P , 310, 311
Wu, J.K., 495
Wu, T.T., 431, 433, 439, 445, 446
Yagi, K., 379
Yamamoto, N , 379
Yamamoto, Y, 275
Yang, H.C., 141, 142
Yang, K.L., 177
Ying, С F, 458
Yoffe, E.H., 322, 338
Yakobori, T , 275, 310
Yonezawa, F., 484
Yoo, M.H., 465
Yoon, H.S , 453, 495
Yoshimura, H., 233
Yu, I.W, 41
Zaoui, A., 495
Zarka, J., 379, 446
Zener, C, 222, 224, 419
Zerna, W., 141, 295
Zhang, H Т., 128, 494, 496
Zhong, W.F., 498
Zienkiewicz, O.C, 275
Zimmerman, R.W., 493
Zorski, H, 348

Subject index
Abel's integral equation, 284
accelerated motion of dislocation, 360
addition theorem, 164
analogy between electromagnetic- and
moving dislocation fields, 361
angular dislocation, 338
anisotropic inclusion, 129
anisotropic precipitates, 220
annealing, 411
arbitrarily shaped plane crack, 297
Asaro and Barnett formula, 337
Asaro et al formula, 334
attenuation, 455
average elastic moduli of composite
materials, 421
average of internal stress, 388
back stress, 402
Bauschinger effect, 402
BCS model, 288, 313
beam, 373
bifuncation, 449
Born approximation, 458
boundary integral equation method, 460
Boussinesq's potential, 489
Brown and Lothe formula, 338
Bulk modulus, 4
Bullough-Gilman crack, 471
Burgers circuit, 45, 68
Burgers vector, 45, 73
catastropic crack propagation, 311
Cauchy principal value, 510
Cauchy singular equation, 475
cavity, 494, 495
center of dilation, 127
Cesaro's integral, 69
chain model, 484
Coffin-Manson law, 379, 385
coherent precipitate, 227
cohesive zone, 281, 472
Colonnetti's theorem, 212
compatibility, 3, 6
complex potential method, 41
compliance, 453, 504
composite material, 213, 421, 433, 453,
492, 493, 494, 495
constitutive equation, 362, 364
continuous distribution of dislocation,
340, 466
Cosserat medium, 53, 439
crack, 240
crack branching, 472
crack closure, 314
crack embryo, 384
crack extension force, 309, 475
crack growth, 307
crack initiation, 379
crack opening displacement, 247, 273
crack tip dislocation, 371
cracks in two phase materials, 471
creep, 412, 447, 449, 494
-function, 450
critical stresses of cracks, 240, 248, 274
582

Subject index
583
cubic crystal, 14, 137, 503
cuboidal inclusion, 104, 121
cuboidal precipitates, 20, 198
cyclic creep, 218
cyclic loading, 314, 379
cylinder, 80, 141, 373
D
debonding, 471, 492, 497
deviatonc strain, 182
deviatonc stress, 362
diffusional relaxation, 406
diffusionless transformation, 359
dilatational eigenstrain, 103, 114, 127, 156
Dirac's delta function, 12, 21, 22, 46, 507
direct observation of dislocation, 44
disclination, 49, 50, 70
direction of-, 50
strain energy of-, 53
-density tensor, 53
-dipole, 338
-loop, 3, 68
-network, 338
-of twist type, 50, 452
-of wedge type, 50
discontinuity, 39, 155
dislocation, 44, 45, 70, 324
direction of-, 45
circular-, 338, 359, 360
-core radius, 354
density tensor, 52, 338
-dipole, 371, 379
-flux tensor, 345, 363
-loop, 3, 68, 324
-pile up, 466, 510
-segment, 328
-velocity tensor, 345
helical-, 338
imperfect-, 45
moving-, 55, 57
partial-, 45
straight-, 327
Volterra's-, 71
dislocation loop density tensor, 339
dispersion curve, 360, 438
dispersion hardened alloys, 398
displacement gradient, 45
dissipation, 361
distant parallelism, 73
distortion, 45, 478
elastic-, 45
plastic-, 45
total-, 45
divergency law, 351
Dugdale-Barenblatt crack, 280, 343
Dundurs' constant, 43, 471, 473
dynamic crack growth, 319
dynamic problems, 497
dynamic fracture toughness, 321
earthquake, 449
edge dislocation, 18
effect of isotropic elastic moduli on stress,
42
effective strain increment, 362
effective stress, 362
eigendistortion, 478
eigenfunction expansion, 458
eigenstrain, 1
-problem, 3
dilatational-, 103, 114, 156
dynamic solution of-, 53
harmonic-, 161
periodic-, 7, 121, 144
polynomial-, 89, 158, 173
shear-, 486, 487
uniform-, 74, 134
eigenstress, 1
eigenvalue, 59
eigenvector, 69
Einstein relation, 418
Einstein summation convention, 499
Eisenstein's theorem, 171
elastic
-compliance, 3, 504
-constant, 3, 504
-distortion, 341, 478
-modulus, 3, 501
-polarization, 1
-stiffness, 504
-strain, 3
-strain energy, 97, 204, 353
ellipsoid, 74

584
Subject index
ellipsoidal cavity, 279
elliptic cylinder, 80, 94, 141
elliptic integral, 83
elliptical plate, 95
embedded weaken zone, 449
embryonic crack, 384
energy and force of dislocation, 353
energy of inclusions, 97
energy of inhomogeneities, 204
energy balance, 215, 319
energy dissipation, 216, 361
energy momentum tensor, 312
energy release rate, 307, 309, 319, 496
England solution, 475
equilibrium condition, 5
equivalency equations, 179, 199
homogeneous-, 199
equivalent eigenstrain, 179
equivalent inclusion method, 40, 178
Eshelby's method, 428
Eshelby's solution, 74
Eshelby's tensor, 77, 443, 448
Euler angle, 423, 445
Euler-Schouten curvature tensor, 73
exterior, 84, 88, 149, 187
extrusion, 379
gamma-function, 165
gange invariant, 353
Gauss' symbol, 174
generalized function, 13
generalized plane problem, 34
Gibbs free energy, 100, 204, 208, 309, 400,
416
grain boundary sliding, 447
granual material, 439
Green's formula, 460
Green's function, 11, 21
-for amsotropic materials, 25
-for transversely isotropic material, 26,
114
-in half-space, 110, 112
-in joint half-spaces, 123
Mindlin's-, 494
-of steady-state, 64
derivatives of-, 32
dynamic-, 55, 57
two-dimensional-, 34
Griffith equation, 286
Griffith fracture criterion, 209, 241, 309
H
f-vector, 458
fatigue, 314, 371, 379
Ferrers and Dyson formula, 95
finite element method, 275
flat ellipsoidal, 83, 143, 161
-crack, 244
flow stress, 362
force constant, 477
Fourier
-series, 9, 505
-integral, 9, 505
fracture criterion, 241
fracture under non-uniform applied stress,
253
Frank dislocation network, 123
Fredholm integral equation, 298
free body, 5
free energy, 100
fundamental equation of elasticity, 3
fundamental metric tensor, 72
half-space, 110, 123, 464, 494
harmonic approximation, 477
harmonic eigenstrain, 161
Hashm-Stnkman's method, 433, 493
heat conductivity, 492
Heaviside step function, 16, 407
Helmholtz equation, 459
Helmholtz free energy, 100
Hencky relation, 367
hexagonal crystal, 14, 139, 503
Hilbert integral equation, 272, 289, 510
Hill's theory, 426
homogeneous equivalency equations, 199
Hooke's law, 3
I
I-integral, 77, 85, 92
image stress, 393
impotent eigenstrain, 199

Subject index
585
-dislocation, 343, 365
-inclusion, 202
inclusion, 2, 38, 74, 129, 177
incoherent precipitate, 226, 414
incompatibility, 1, 7, 65
-tensor, 69
Indenbom and Orlov formula, 48
indenter, 368
inhomogeneity, 2, 40, 177, 181
anisotropic-, 187
inhomogeneous-, 177, 180, 452
-inclusion, 179
instability, 449
interacting tension cracks, 276
interaction between dislocations and in-
inclusions, 463
interaction energy, 99, 208, 353
interface discontinuity, 39
interior, 75, 92, 133
internal friction, 419
internal stress, 1
intrusion, 379
irreversibility of plastic work, 355
isotropic, 13, 22
transversely-, 14, 26
J-integral, 311
jointed half-space, 123
К
Kelvin's material, 218
kinematically admissible field, 373
Kronecker delta, 2, 481, 499
Kroner's expression, 340
Kroner's formula, 31
line integral expression
-for displacement, 348
-for strain, 47, 48
-for velocity field, 349
-plastic strain, 348
line of discontinuity, 367
linear connection, 73
liquid crystal, 49
locked dislocation, 467
Lorentz force, 357, 360
-condition, 353
-contraction, 359
M
Mach cone, 359
macroscopic strain, 389
martensite, 218, 492, 493
spherical-, 234
-transformation, 229
mass diffusion, 228
Maxwell tensor of elasticity, 311
Mellin transform, 471
Mindlin's solution, 112
minimum principle for strain, 441, 442
minimum principle of free energy, 228
minimum strain energy, 223
Mises yield criterion, 362, 364, 365
misfit strain, 1, 406
modulated structure, 224
Mori and Tanaka's theory, 394, 453, 480,
492
motor, 53
Mott assumption, 379
moving dislocation, 55, 57, 355
Mura's formula, 45, 47
N
Laguerre polynomial, 467
Lame constant, 4
Laplace transform, 508
lattice theory, 477
Legendre polynomial, 31, 161
-function, 489
Levy and Mises equation, 303
Newtonian viscous solid, 218
non-elastic matrix, 492, 494
nonelastic strain, 1
non-metallic inclusion, 238
non-uniform applied stress on a crack,
268
Nye's tensor, 53

586
Subject index
О
oblate spheroid, 84, 94, 488
Orowan mechanism, 398
Orowan-Irwin formula, 286
-theory, 310
Orowan's loop, 342, 399, 404, 408
orthorhombic material, 142, 504
oscillating dislocation, 359
Pankovich-Neuber function, 124, 282
parallel dislocation, 325
Pans law, 260, 314, 317
path-independent integrals, 313
Peach-Koehler force, 313, 355, 463, 466
Peierls dislocation, 358
-stress law, 360
penny shape, 81, 184, 217
-crack, 240, 252, 292, 318
periodic eigenstrain, 7, 121, 144
-distribution of spherical
inclusions, 165
permutation tensor, 6, 8, 499
persistent slip band, 379
Petch-type equation, 379, 387
phase transformation, 1
plane strain, 5, 25, 275
plane strain problem, 365
plane stress, 5, 25, 275
plastic distortion, 339, 341
-rate, 346
plastic polycrystal, 439
plastic rotation, 51
plasticity, 361
Poisson's ratio, 4
polarization, 1
polarization stress, 434
polycrystal, 439
polygonization, 344
polynomial applied strain, 188
-eigenstrain, 89
-eigenstrains in anisotropic materials,
158
porous media, 439
potent dislocation, 365
potential energy, 99, 479
Prandtle-Reuss relation, 361, 363
precipitation, 180, 218, 492, 493
process zone, 310
projection integral equation, 496
prolate spheroid, 84, 488
П integrals, 255
R
Rj-conjunction, 303
ratcheting, 379
Rayleigh-Gauss approximation, 458
Rayleigh wave, 322, 358
recovery strain, 412
reduced strain, 182, 453, 502
-stress, 362, 502
relaxation, 406
-function, 450
-moduli, 453
-strength, 419
-time, 418
residual stress, 1
resolved shear stress, 440
Reuss approximation, 424
Reuss equation, 363, 364
Ricci tensor, 71
Riemann-Christoffel tensor, 71
Riemannian space, 71, 72
rod, 185, 217
Rodrigues* formula, 164
Rudmcki's theory, 449
scattering cross-section, 359, 498
Schouten's torsion tensor, 73
screw dislocation, 15
moving-, 57
seismic velocity, 455
self-consistent method, 430, 443
self-force, 354
sgn ?, 507
shear line, 366
shear modulus, 4
singular, 192, 198
sink of dislocations, 53
sliding inclusion, 484, 492, 497
slip, 2

Subject index
587
-plane, 45, 440
slipping interface, 465
slit-like crack, 242, 271
Somigliana dislocation, 48, 485, 497
-equation, 298
source of dislocation, 53
sphere, 79, 93, 183
spheriod, 84, 137, 223, 488
spherulite, 498
standard linear solids, 419, 453
state quantity, 51, 350
Stokes' theorem, 47
strain
elastic-, 46
total-, 46
strain energy, 97, 204
-of dilatational eigenstrain, 103
-of elliptic cylinder, 101
-of penny-shaped flat ellipsoid, 101
-of spherical inclusion, 101
-of spheroid, 102
strain hardening, 441
strain polarization, 436
stress concentration factor, 40, 155, 277
stress corrosion, 387
stress-free transformation strain, 1
stress intensity factor, 155, 245, 260, 264,
275, 297, 496
elastodynamic-, 320
stress intensity function, 320
stress jump, 40
stress orienting precipitation, 237
stress singularity, 323, 471
oscillatory-, 472, 475
surface dislocation density, 341, 367, 373
Tanaka-Mori's theorem, 390, 429
Taylor's analysis, 439
Taylor's hardening law, 443
tetragonal crystal, 504
thermal crack, 276
thermal expansion, 1, 88, 492
-coefficient, 2
thermal stress, 1, 88
thermoelastic effect, 360, 493, 494
thin film, 123
threshold velocity, 358
torsion, 192, 376
total strain, 3
-potential energy, 99, 208
-toughening, 495
transformation strain, 1
transversely isotropic, 26, 503
Tresca's yield condition, 293
twin, 230, 234
two ellipsoidal inhomogeneities, 192
two gas bubbles, 198
two spherical cavities, 196
U
uniform eigenstrain, 74, 134
upper and lower bounds, 433
V-integral, 91
variant, 237
velocity intensity function, 320
viscoelasticity, 449
Voigt approximation, 421
Voigt constants, 140, 502
Volterra formula, 45
W
wave in viscoelasticity material, 453, 455
wave equation, 362
wave scattering, 455
wave velocity
dilatational-, 56, 455
shear-, 56, 455
Weibul distribution, 495
Wiener-Hopf technique, 471
William's coefficient, 275
Willis' formula, 311
Wnuk's equation, 311
work-hardening, 362, 398, 404
Y
yield function, 447
yield stress, 362
yield surface, 446
Young's modulus, 4, 501
Zener arusotropic factor, 222
Zener-Stroh crack, 471