Dislocations in Solids: Elastic Theory v. 1 (Dislocations in Solids)

Solids Volume 1 Elastic Theory

Edited by

R. N. NABARRO Department ofPhysics University of the Witwatersrand Johannesburg, South Africa

1979

North-Holland Publishing Company Amsterdam- New York- Oxford

© NORTH-HOLLAND PUBLISHING COMPANY, 1979 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, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

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Library of Congress Cataloging in Main entry under title:

Data

Dislocations in solids. Bibliography. Includes index. CONTENTS: v. 1. The elastic theory. 1. Dislocations in crystals - Collected works. 2. Solid state physics Collected works. I. Nabarro, Frank Reginald Nunes, 1916QD921.D55 548'.842 78-10507 ISBN 0-7204-0756-7

Printed in Great Britain

The study of dislocations in solids has come of age. Our basic knowledge has expanded beyond the scope of a single volume, and we devote five volumes to the fundamental properties of dislocations and their influence on the properties of solids. In this first volume, J. Friedel outlines the whole field from a modern physical standpoint, A. M. Kosevich gives the theory of dislocations in an elastic continuum, J. W. Steeds and J. R. Willis extend this theory to anisotropic media, and J. D. Eshelby considers the interaction of dislocations with boundaries. Finally, B. K. Datta Gairola tackles the formidable problems which arise when we go beyond the approximations of linear elasticity, as the nature of the subject requires. Having painted the general picture, we plan to extend the series with a volume on the practical applications of dislocations, and finally a survey of recent developments. I cannot end this preface without thanking Mrs. O. L. Prior, who not only ensured some consistency of style and notation within and between the chapters, but also detected errors in the work of my most distinguished colleagues. The task of editing was greatly eased by Mrs. A. B. Alexander, my secretary for many years. F. R. N. Nabarro

Volume I

Preface v Contents vii

J. Friedel In troduction 1. A. M. Kosevich Crystal dislocations and the theory of elasticity 2. J. W. Steeds and J. R. Willis Dislocations in anisotropic media 3. J. D. Eshelby Boundary problems

167

4. B. K. D. Gairola Nonlinear elastic problems Author index 343 Subject index 347

223

143

33

/

Dislocations Introduction J. FRIEDEL Laboratoire de Physique des Solides Unioersite de Paris-Sud, 91405 Orsay, France

© North-Holland Publishing Company, 1979

Dislocations in Solids Edited by F. R. N. Nabarro

Introduction Introduction

3

1. Dislocations as singularities of an order parameter 3 1.1. Volterra process for continuous homogeneous and isotropic (classical) elastic media

3

1.2. Volterra's process in a continuous, homogeneous but anisotropic solid medium. Perfect and imperfect dislocations 8 1.3. Volterra's process for crystals

10

1.4. Burgers circuit for elastic solids

12

1.5. Extension of the notion of Burgers circuit for rotation dislocations. Disclinations 1.6. Possible dislocations in some molecular crystals 14 1.7. Singularities in liquid crystals 16 1.8. Singularities in other order parameters [Toulouse and Kleman] 2. Cores of translation dislocations in crystals 20 2.1. Core cut-off [Volterra] 20 2.2. Dislocations as solitons [Dehlinger, Peierls]

20

2.3. Splitting of the core [Heidenreich and Shockley] 2.4. Atomic description of the core 23 3. Dislocation ensembles 24 3.1. Dislocation networks in strained f.c.c. metals 3.2. Dislocation networks in recovered f.c.c. metals 3.3. Deformation and recovery in other structures 3.4. Deformation of polycrystals 25 3.5. Inhomogeneities in deformation 4. New techniques for old problems 5. More complex materials 27

26

26

6. Physical properties other than plasticity References 31

18

13

Reviewing the present state of the art, one is struck not only by a definite change in climate in the last ten years, and by a renewal of activity extending to newfields, but also by the permanence of some fundamental trends and underlying problems. In the new activities, one must count both the use of new techniques, which have helped to solve old problems, and also new fields of research: materials more complex than the simple metals of classical metallurgy, or physical properties other than plasticity. In these new activities, one finds the same complementarity between a macroscopic linearized description of long-distance distortions and an atomic description of the dislocation cores. Which aspect dominates a given physical property is still not always clear. But the major problem which is still not completely solved is that of bridging the gap between the properties of individual dislocations and the behaviour of dislocations "en masse", as observed in strongly deformed materials. We shall comment on these various aspects in turn.

22

1. Dislocations as singularities of an order parameter

24 25 25

We first focus our attention on the long-range distortions around dislocations. These distortions are small enough to be analyzed by linearized equations, e.g. the linear elasticity for solids. Dislocations appear as singularities of distortions. After comparing the standard Volterra construction and Burgers circuit characterization for continuous solids and crystals, we analyze their extension to other media such as plastic crystals and liquid crystals, and the relations of dislocations with other singularities of an order parameter.

30

1.1. Volterra process for continuous homogeneous and isotropic (classical) elastic media

The state of strain in such a medium can be defined by the spatial derivatives of the displacement vector u of each point I' from an assumed unstrained state ofthe medium. The corresponding state of stresses 0'(1') is obtained by Hooke's law, i.e. by the elastic constants which relate linearly the components of 0' to the distortions such as -!(ou)ox j + ouj/ox i ) . In the media discussed here, the elastic constants have the same value everywhere and reduce to two independent constants. An un strained medium has no stresses and strains, i.e. along an internal surface S, the two parts of the medium exert no force on each other; thus when a cut is made along S, the two lips S', S" of the cut have no tendency to move with respect to each other (fig. I). Some media are contrariwise internally strained and stressed in the absence of externally applied stresses. Volterra's construction was introduced to analyze such states ofinternal stresses as due to a continuous distribution ofdislocation

J. Friedel

4

Dislocations

an introduction

5

lines of infinitesimal strength. It is worth recalling the reasonings in this historical case before extending the results to more complex cases. Volterra's process is (fig. 1):

d

to cut the medium along a surface S limited by a line L. S', S" of the cut with respect to each other, without distorting them. y - to fill any empty space thus produced by (unstrained) material, or conversely to remove any extra matter possibly produced in this gedanken operation. (J 0 and F res = constant, then the dislocation source may produce, in principle, an unlimited succession of closed dislocation loops. However, often the quantity F res grows with the increase of the radius of the loop expanding in the glide plane. If the retarding force F res increases proportionally to the radius of the loop, then the dislocation source may produce a limited number of dislocation loops. This case was studied by Van Bueren [33]. Orlov [34J discussed the operation of the Frank-Read source in another case when the resistive force grows with the increase of the radius of the loop but this growth saturates. The Frank-Read source can be formed by the way of so-called double cross slip. The deviation of a screw dislocation from the glide plane is known as cross slip (see

80

A. M. Kosevich

Ch.1

sect. 4.1.). Consider a dislocation in which a screw segment (AB in fig. 34) cross slips, forming a loop AA'B'B in the cross slip plane. The segment A'B', which is a pure screw one, can move by double cross slip into a new glide plane parallel to the original slip plane. If the points A' and B' are fixed, the segment A'B' acts as a Frank-Read source.

Crystal dislocations and the theory ofelasticity

81

5.7. The dislocation model of the twin The simplest example of a twin in the crystal is schematically shown in fig. 36. The atomic layers on the opposite sides of the so-called twin plane ZO X are turned relative to each other by an angle 20:. The twinned and un twinned parts of the crystal are mirror reflections of each other in the plane mentioned. The interface boundary between the upper and lower parts of the crystal in fig. 36 is a perfect lattice plane; in that case one speaks of a coherent twin boundary. y

Fig. 34 Double cross slip. A screw segment of dislocation AB moves out of the primary slip plane into the cross slip plane. A screw segment A'B' of the loop in the cross slip plane moves into a new glide plane parallel to the primary slip plane.

If the dislocation lying in its glide plane is fixed only at one but not two points, it can also serve as a dislocation source. It is said in this case that a pole mechanism of the generation of dislocations operates. When all parts of the dislocation line move across the glide plane with about the same linear velocity, the dislocation line winds itself into a spiral in the glide plane. The successive positions of a dislocation fixed at a point A are shown in fig. 35. Such a source does not emit a succession of dislocation loops but a dislocation spiral. The corresponding pole mechanism is often called a mill.

Fig. 36 A coherent twin boundary. The angle of twinning is 2a.

If the twin has finite transverse dimensions then the outline of its cross section in the x-y-plane is a curve (fig. 37) and therefore it cannot coincide with a crystal plane. Vladimirskiy [36J has shown that, in the general case, the twin boundary consists of single coherent areas separated by twinning dislocations (fig. 38). The Burgers vector of such a dislocation lies in the twinning plane, which is simultaneously the glide plane of the twinning dislocation. The twinning translation is always smaller than the slip translation.

------5 ~-----4 ~-----3

_------2

Fig. 35 The pole mechanism of the generation of dislocations. A dislocation is fixed at point A. Under stress it takes successively the configurations I, 2, ... ,5, forming a spiral source with A as a centre.

A mechanism which has much in common with the Frank-Read source described above may also operate when the dislocation moves by climb instead of glide. This type of source is usually called a Bardeen-Herring source [35J.

Fig. 37 The cross-section of a twin at the surface of a crystal.

Fig. 38 A twinning dislocation after Vladimirskiy [36J.

The twinning dislocations are located along the contour of the twin (fig. 39). The density of such dislocations is determined by the curvature of the boundary of the twin.

82

A. M. Koseuich

Ch.1

Before discussing the dislocation description of the twin it is useful to consider a twinning dislocation from another point of view. Let us imagine the twin band shown in fig. 40. The monatomic edge of this band is a partial Shockley dislocation with its Burgers vector b = 2a tg (J.. where a is the distance between the crystal planes in the direction perpendicular to the twinning plane, and 2(J.. is the angle of twinning.

83

Crystal dislocations and the theory of elasticity

§5.7

Let us consider an elastic twin of infinite length and uniform in the z-direction and in a plane stress field Pik(X, y); this is a two-dimensional problem of elasticity theory. Athin twin in this problem is equivalent to an array of straight twinning dislocations lying along the z-axis and distributed along the x-axis. Further, we assume that the dislocations are distributed continuously along the x-axis. The dislocation theory of twins based on this model is given in detail in the survey by Kosevich and Boiko [37J. Here we only write and discuss some basic equations of this theory*.

~"X

~L (a)

~(x)

x

1 1.. 11.

-L

D

L

( b)

Fig. 42 A twin within the crystal. (a) A cross section of the twin lamella. (b) A schematical diagram of the twinning dislocation density f0(x). The total strength of all dislocations is zero. Fig. 39 A section of the twin boundary with twinning dislocations.

Fig. 40 A monatomic twin lamella ends on a partial Shockley dislocation (shaded circle).

The macroscopic twin is a set of the monatomic planar stacking faults described above which terminate in its boundary. Therefore the twin sketched in fig. 37 may be replaced by the array of twinning dislocations shown diagramatically in fig. 41. In this section we shall analyze only the so-called elastic twin*, the end of which is not fixed. The thickness of the elastic twin h is generally very small as compared with its length L, i.e. h « L, and according to Vladimirskiy [36J we may consider all dislocations to be situated in the same glide plane or twinning plane. Thus the twin may be considered as a certain pile-up of twinning dislocations. Therefore the problem ofthe equilibrium of a twin in an elastic field is the problem ofthe equilibrium of a corresponding dislocation pile-up.

For simplicity, we consider the twin to be not near the surface but within the crystal (fig. 42a). Let us assume that at the origin (x = 0) there is a source of twinning dislocations which can generate under the applied load any number of pairs of straight dislocations with opposite signs. We introduce the density of twinning dislocations £0(x) which is evidently related with the thickness ofthe twin hex) at the point x by "L

hex)

= a

jx

£0(x') dx'.

(101)

We suppose that the stresses are symmetrical about the centre of the cross section of the twin. Then the outline of the twin will also be symmetrical and hex) = h( -x), but £0(-x) = -£0(x) (see fig. 42b). The equation of equilibrium of the dislocations along the twin can be written similarly to eq. (75) as '' L

j

-L

£0(x') dx' _ , - w(x). X

-

(102)

X

However the external forces on the dislocations which are described by the function w(x) must be discussed specially. ----L

Fig. 41 A dislocation model of the macroscopic twin at the surface. The length of the twin is L and the thickness is h.

* If a twin lamella appears when the load is applied and vanishes when the load is removed, elastic twinning is said to take place. If the twin lamella remains after stress relief it is called a residual twin.

We should bear in mind that the slowly moving dislocation is acted upon by forces of inelastic origin (see sect. 5.5.). Firstly, there is a force analogous to some degree to the "dry friction" force, of the type of the Peierls-Nabarro force. Secondly, there is the force of the surface tension on the twin boundary. It is evident that the latter force acts only on the dislocations located at the end of the twin. In fact, the addition of one dislocation at any part of the twin where its width has macroscopic dimensions in practice hardly changes the area of the interface surface between the twin and the matrix and does not essentially change the surface energy. At the same time, the * See Addendum 5 and 8.3.

84

A. M. Kosevich

Ch.l

addition of one dislocation near the end of the twin, where the interface boundaries are separated by a few atomic parameters may essentially change the corresponding surface energy. This statement is confirmed by numerical calculations [38]. The difference in the character of the crystal distortions caused by the dislocations at the end of the twin (by the head dislocations of the pile-up) and by the dislocations on the interface surface can be seen by comparing the diagram in fig. 40 which shows the leading Shockley partial and the diagram in fig. 38 which shows the Vladimirskiy twinning dislocation. Write the function w(x) in the form

Crystal dislocations and the theory of elasticity

§5.7

We have replaced the integrals from - L to L by the integrals from 0 to L, using the symmetry of the problem. Since ST(X) is non-zero only in the region L - x rv d « L wecan put Lf - x 2 :::::; 2L(L - x) in the second integral on therightofeq. (107) and substitute eq. (105). The condition of eq. (107) then takes the form

2

r

a(x) dx

L

~Jo (L 2

= a(x) + S(x\ D

(103)

D

where a(x) = p~/x) and D = fJh/2n(1 - v) in the isotropic approximation. The function Sex) describes the forces of inelastic origin. If the external stresses a(x) grow and the twin increases its size, we have Sex)

= - So - ST(x) ,

(104)

where the minus sign determines the direction of the forces of inelastic origin in the present case. Here So is the dry friction force and ST(X) is the surface tension force. The sign of the force ST(X) is always the same, whereas the sign of the dry friction force varies depending on the direction of the dislocation movement. The possibility ofchanging the sign ofthe dry friction force is the reason for the hysteresis in twinning. The force ST(X) depends on the distance hex) between the twin boundaries and is non-zero only at small hex), i.e. in the narrow region d near the end of the twin. It is natural to suppose that the shape of the twin near its end is determined by the nature of the surface tension and does not depend on the applied external loads. Then in finding the shape of the main part of the twin, the quantity ST(X) becomes a given function independent of o (x):

S (x) = T

{geL - x), geL + x),

x > 0; x < o.

(105)

The definition of eq. (105) is valid only over the region d outside which this function is unimportant. Since the ends of the twin are assumed not fixed, the stresses on them must remain finite. This means that in solving the integral equation (102) we deal with the case for which the solution is obtained by using eq. (78). For our choice of the origin at the midpoint of the segment ( - L, L) this formula becomes .@(x)= -

~ (L 2

X 2 ) 1/ 2

-

n?

pJ"L -L

(L 2

w(x') -

X'2)1 /2

~. x' -

(106)

X

X 2)1/2

-

rg

= So + L 1 / 2 '

(108)

where rg denotes the following constant

_ 2 w(x)

85

rg -

1/2

fOO g(~) d(

tt

0

~ 1/2

(109)

A constant like rg arises also in the dislocation theory of cracks (see, for example, [I4J and the original papers by Barenblatt [39J). This constant can be expressed in terms of the ordinary macroscopic properties of the crystal, i.e. its elastic modulus, and the surface tension on the twin boundary [37J. If one is interested in the properties of the twin at points far away from its ends, then the function g(~) can be replaced by the following delta-function: (110) Relationships of the types given by eqs. (106) and (108) allow us to present a total description of elastic twins in the crystal.

6. Systems ofstationary dislocations in a crystal 6.1. Continuous distributions of dislocations Returning to the initial definition of the dislocation of eq. (1), we rewrite it in the form

ls~

dxiu ik

=

-b k ,

U' k I

=

OUk.

ox,

(111)

The tensor U ik in eq. (111) is often called the simple shear. We shall call U i k the elastic distortion tensor. Its symmetrical part yields the ordinary elastic strain tensor (112) In addition to the condition of eq. (1), which determines the type of the dislocation singularity, it is assumed [1, 40J that in the presence of a dislocation in the medium the elastic distortion tensor U ik is a one-valuedfunction of the coordinates, continuous and differentiable over all of space", In dislocation theory, the distortion tensor is usually regarded as an independent quantity describing the deformation ofthe crystal.

The requirement of eq. (79) must be satisfied, and it gives in this case

L

fo (L

a(x) dx 2 -

X 2)1 /2

1

= 2 nS O +

fL 0

ST(X) dx

(L 2

_

X 2)1/2

(107)

* If we confine ourselves to the requirement that the strain tensor eik and not the distortion tensor U ik be unambiguous, continuous, and differentiable, then we can take into consideration also dislocations of more general type than those defined by the property of eq. (1) (see, for example, [1]).

86

A. M. Koseoich

Ch.l

We write the condition of eq. (111) in a differential form. To do so, we transform the integral round the contour s into one over a surface I spanning this contour:

,h t

=

dxiu ik

f

dIil\/m

oU mk ox/ .

The vector b can be written as an integral over the same surface by means of the twodimensional delta-function b,

~

f

fA b(~) az,

ox,

= - t.b,

s: ) u(~ .

(114)

This is the required differential form. It is clear that on the dislocation line itself (~ ~ 0) which is a line of singularities, the representation of the distortion tensor as the derivatives U ik = OUk/OX i is no longer meaningful. If a crystal contains a large number of dislocations which are at relatively short distances apart (although far apart compared with the lattice period, of course), it is useful to treat them by means of an averaging process. We apply this consideration in those problems in which we are not interested in the exact distribution of the field between the different dislocations and in which the theory operates with physical quantities that are averaged over small elements of volume. It is clear that rather many dislocation lines must cross such "a physically infinitesimally small" volume. An equation which expresses a fundamental property of dislocation deformations can be formulated by a natural generalisation of eq. (114). We define the dislocation density tensor CX ik in such a way that its integral over a surface spanning any contour s is equal to the sum of the Burgers vectors of all dislocation lines embraced by the contour:

f~;k

dE,

The tensor

CX ik

= b.,

87

the index f3 runs over the possible directions of the vector b, and pp(t, r) is the density of the distribution of the vectors band t over the possible directions, r) dO being the number of dislocations having a Burgers vector direction f3 V"~''-'U''b through a unit area perpendicular to the vector t, and located inside a solid angle dO around the direction of t. The integration is carried out over the complete solid angle. seen from eq. (116) the tensor CX ik should satisfy the condition (118)

(113)

where ~ is the two-dimensional radius vector taken from the axis of the dislocation in the plane perpendicular to the vector t at the given point. Since the contour s is arbitrary, the integrals can be equal only if the integrands are equal: oUmk Gilm -;:)-

Crystal dislocations and the theory of elasticity

in the case of a single dislocation states simply that the Burgers vector is along the dislocation line. When the dislocations are treated in this way the tensor U ik becomes a primary quantity describing the deformation and determining the strain tensor through (112). A displacement vector u, related to U ik by the expression U ik = OUk/OXi cannot exist. Really, this is clear from the fact that with such a definition the left-hand side of eq. (116) should be identically zero throughout the crystal. Equation (116) together with ""V11"~''''U~

0PidoXi

= 0

(119)

and with Hooke's law form a set of equations of equilibrium for the continuous distribution of dislocations in the medium [1, 7, 40]. The tensor of the Burgers vector density IX ik in eq. (116) should be considered as a certain continuous function of the coordinates, satisfying the condition of eq. (118). Sometimes, instead of eq. (116) which defines the elastic distortion tensor, one uses an equation for the derivatives of the elastic strain tensor eik . This equation may be obtained by applying the operator Gjpk % x p to both parts of eq. (116) and symmetrizing the result over the indices i and j. Then we have

02

e mk Gilm Gjpk ox/ oX

(120) p

where

(115)

(121)

now replaces the expression on the right of eq. (114) (116)

and describes the continuous distribution of dislocations in the crystal. If we now take into account the fact that the vector b can have only a fixed number of fully defined directions in the crystal then the average dislocation density CX ik can in this case be written in the form [41, 42J

~,.(r) = ~

f

fibfpP(t, r) dO,

(117)

The tensor Yfik is the so-called Kroner's incompatibility tensor [7]. The appellation of the tensor comes from the fact that if Yfik = 0, then eq. (120) coincides with de SaintVenant's compatibility equation for the components of the strain tensor. It follows from eq. (121) that 0YfidoX i

= O.

(122)

This is the continuity equation which states that the incompatibility is solenoidal. Equations (120), (119) and Hooke's law also make a set of equations of equilibrium of the medium with continuously distributed dislocations. There are many ways of solving this system [1, 7, 41]. However, the introduction of the tensor Yfik seems

Ch.l

A. M. Koseoich

88

inconvenient in the analysis of moving dislocations. Therefore we shall refer in further discussion mainly to eq. (116) and characterize the dislocation distribution by the tensor (Xik •

Crystal dislocations and the theory ojelasticity

&"'n .... ,n.c~Tl

The simplest but frequently occurring case of a dislocation distribution is that when the total Burgers vector of all dislocations (denoted by B) is zero. The presence of a dislocation involves a certain bending of the crystal as sketched in fig. 43a (greatly exaggerated). It is clear that in order to produce the macroscopic bend of the crystal, the latter should contain a macroscopic number of dislocations of the same type. It is readily seen that if there are a large number of uniformly distributed parallel edge dislocations of the same sign in the crystal, then this crystal is bent with the curvature I/R = BjS, where B is the total Burgers vector of the dislocations, R is the radius of curvature and S is the cross-sectional area of the bent crystal (see fig. 43b).

the deformed crystal. The total dislocation moment Dik of the crystal is, by of eq. (16),

o.; = ~ s.s,

6.2. Dislocation polarization of the deformed crystal

89

'" 1811mL

'" ±4: silmbk fX

I

dx;

f, mb, dl ~ ! ISilm t

(125)

x, IX"" dr,

the summation is over all dislocation loops and the integration is over the "1"11"" of the crystal. Substituting eq, (124) in eq. (125) we obtain

'"

f »; '" I 1),. =

!

SilmSm,qXI

~~:k dr '" !

f

xm(O£;, -

~~:) dr

after integration by parts in each term,

Pi' dr.

(126)

It follows from eq. (126) that the tensor P i k is actually equal to the dislocation moment of unit volume of the body. Apparently this tensor was introduced for the time by Kroupa [43]. We shall call P i k the dislocation polarization tensor. substitute eq, (124) in eq. (116) and rewrite the latter in the form

(a)

(127) (b) Fig. 43 A plastically bent crystal. (a) Crystal bending caused by one edge dislocation. (b) The crystal is bent with curvature 1/R. It contains a macroscopic density of parallel edge dislocations.

And so the equality B = 0 means that there is no macroscopic bending of the crystal as a whole. The condition B = 0 is, for example, a property of a system of closed dislocation loops. This condition signifies that integration over any cross section of the body gives

I

rJ. ik

dl: i = O.

(123)

From this it follows that the dislocation density may be represented as (Xik

=

ee.;

Cilm

-8-'

(128) It follows from eq. (127) that the tensor of the derivative of some vector Wi k

"""

8uk/8x i ·

W ik

may be always represented in the form (129)

We shall measure the crystal deformation from the state when P i k = 0, assuming that the entire deformation process occurs with B = 0. Then the vector u can be set to the vector of the total geometric displacement u T in the crystal with dislocations, and the tenser W i k determines the total distortion tensor of the deformed body. The equation for the vector u T is derived by using Hooke's law and by substituting eqs. (128) and (129) in eq, (119). If external body forces are absent, this equation reduces to

(124)

Xl

where P i k is a tensor which is naturally zero outside the body. Then the integral of eq. (123) transforms into the integral along a contour outside the body, and becomes zero. It should also be noted that eq. (124) necessarily satisfies the condition of eq. (118). It is easy to see that the tensor P i k thus defined represents the dislocation moment

Thus, when the dislocation distribution is known and the dislocation polarization tensor P ik of the body is specified, the vector u T can be determined from the static equations of elasticity theory, in which the body forces have the density (130)

A. M. Koseoich

90

Ch.1

Crystal dislocations and the theory of elasticity

It should be noted that the force of eq. (130) is in natural accordance with eq. (9) for the force J;(S) which was used for finding the geometric displacement vector around a single dislocation loop. In order to see this we consider a large number of dislocation loops and average eq. (9) over unit volume. The integral Je~~) dr where e~~) is defined by the formula of eq. (5) or the equivalent expression of eq. (43), is equal to

solution of this equation which vanishes at infinity is

Je~1) dr

=

i

J(n.b, + nkb,J az.;

(131)

The right-hand side of the relationship of eq. (131) may be simplified using the notation of eq. (16)

Je~1) dr

= i(dik

Let us consider the equations of equilibrium for an isotropic medium in the case when body forces are absent and the stresses satisfy eq. (119). Equation (119) is satisfied identically if we write (132)

where Xik is a symmetric tensor called the stress function. Kroner [7J introduced a different stress function X;k which depends on the elastic constants of the isotropic medium and is defined by the relation

(133)

(134)

The relation between strain and stress in the isotropic body is given by

eik =

2~(P" -

1 : v Pll Oik}

L1 ==

o2joxt·

+ C,klm dlJ,

R =

Jr - r'l.

(138)

Finally we can find the stress, which is given by

i 1 (02 +~ oX a~k i

- 6ik L1X;[

)J

.

Kunin [44J has obtained by a different method the stresses in an anisotropic medium of any symmetry when 1Jik is given.

Let us consider a system of single dislocation loops in an infinite isotropic elastic medium. The energy of the elastic field of the strains eik and stresses Pik of this system is determined by the standard formula V =

±JeikP" dr.

(139)

We use the representation of the tensor Pik in terms of the stress function Xik and substitute eq. (132) into eq. (139) V --

-21

JeikC,ilmC,kpq-a;:) 02 Xmq dr. ».

(140)

Integrate by parts in eq. (140) and omit the surface integrals at infinity. Then the integral of eq. (140) may be transformed into the form V =

±J~ikXik dr= f1 J(~ikX:k + 1 ~ ~iiX;') dr. V

(141)

In writing down the second part of the equality of eq. (141) we used eq. (133). Now, we substitute eq. (138) into eq. (141) and obtain (135)

It follows from eqs. (120) and (132)-(135) that the tensor X;k satisfies the equation 'L1L1X;k = IJik'

J

, . 1 a R( C,ilm dlk Xik(r) = 16n b, 8X m

uX p

We can show that it is permissible to introduce the three subsidiary conditions

0X;kjOX i = O.

(137)

Interaction between dislocation loops

6.3. Kroner's stress function for an isotropic medium

~ 2f1( X:k + I ~ v X:, Oik}

- r'l1Jik(r') dr,

The stress function of eq. (137) for an isolated dislocation loop reduces to the form

Pu. = 2p [ L1X;k

+ d,,),

where dik is the dislocation moment of a single loop. The subsequent summation over the dislocation loops in unit volume makes evident the correspondence between j(S) and eq. (130).

Xik

-~ fir 8n

X;k(r) =

91

(136)

This is Kroner's basis relation between his stress function and the incompatibility tensor, subject to the subsidiary conditions of eq. (134). The conditions of eq. (134) are satisfied because IJik satisfies eq. (122).

V = - :"

JJ~ik)lf

-

f'I~,,(f') dr dr

8"(;~ v) JJ~i,(r)lf - f'l~kk(f') dr dr.

(142)

Substituting eq. (121) and using a partial integration we can express the energy V terms of the dislocation density O'.ik' We shall not give this relation here, but note

A. M. Kosevich

92

Ch.l

that it is now possible to write the elastic energy of the system of single dislocations in terms of line integrals along the dislocation loops: V --

P "LaPbab ik i k »

1

Crystal dislocations and the theory ofelasticity

dlS:1oc;atllon is made up of a continuous distribution of infinitesimal dislocations. Let distribution be represented by the density

(143)

2" i...J

a,p

(146)

where a and f3 are the dislocation loop numbers and summation is over all dislocation loops. The quantities Lft are given by

(144)

where ~ is a two-dimensional vector measured from the dislocation axis in a plane perpendicular to t, and y(~) is some function that differs from zero in a small vicinity of the dislocation line. If we use eq. (146) in the calculation of the coefficient of the dislocation selfinductance L i k then the expression for L ik becomes

L'k

where

R = IY a

-

Ypl,

n = (Y a

-

_

1

~ 41C(t_ v) f f y(~)y(n d2~ d2~'

Yp)/R.

Other possible expressions for the quantities Lft are given in the survey by de Wit [45J, where the coefficients Lft for loops of simple geometrical shapes are written. These results represented by eqs. (143) and (144) were first obtained by Kroner [7]. It is cenventent to divide the sum of eq. (143) into two sums

-

93

"'LaPbabP ik i k»

2" L,;"

a*p

(145)

The first term, Vin t in eq. (145) is the interaction energy of the dislocation loops. Formally, it looks like the expression for the energy of magnetic interaction in a system of linear, currents. The coefficients L~t are called by Kroner the coefficients of the "dislocationk'mutual inductance" [7]. The-second term in eq. (145), namely Vself , is the sum of the dislocation selfenergies;. Therefore it is natural to call the coefficients Lf: the coefficients of the "dislocation self-inductance". The detailed discussion of the self-energy is given in the next section.

6.5. Tile self-energy of a dislocation loop Let us consider an isolated dislocation loop of arbitrary form in an isotropic medium. Its self-energy is.determined by the coefficient of self-inductance Lf:, which can be found by using the same curve for l'' and IP in eq. (144). Thus we must integrate over the same curve twice in the foregoing expressions for self-energy. It is known from magnetostatics that the coefficient of self-inductance of a line wire having zero cross section makes no sense, as the corresponding double integral diverges. This situation is analogous to that which occurs in the case of a line dislocation in the theory of elasticity, namely, the coefficient L~: for the dislocation having zero cross section is not meaningful. To obtain a finite self-energy we assume that the dislocation is not infinitesimally thin but has some small thickness, i.e, that the single

x

f f R(l: I') [(0" + n,n

k)

dl

ar -

(1

+

v) dl, dlkl

(147)

Let us use eq. (147) and write the self-energy of the dislocation loop in the form

v, ~

ff

B(l, l') dl dl',

(148)

where 8(1, I') has the meaning of a non-local density of the dislocation self-energy. An expression for 8(1, I') follows directly from the formula of eq. (147) 8(1,1') = {[b 2

+ (bn) 2 Jtt' f

Jl

X

8n(1 - v)

- (1

+

v)(bt)(bt')}

fy(~)y(~/) d2~ d2~' R(I, I')

.

(149)

We note that the vector n is taken outside the sign of integration with respect to in eq. (149). Allowance for n in such an integration leads to a small correction to eq. (148) if the ratio of the "thickness" of the dislocation line to the characteristic linear dimension of the entire loop is small. The latter ratio will be assumed to be very small. We can introduce a quantity which has the meaning of the self-energy per unit length of dislocation. Integrate eq. (148) with respect to one of the variables over the loop and write ~ and ~'

v, ~

T

U(I) dl,

U(l)

=

TB(I, I') dl'.

(150)

It is important that the self-energy per unit length of dislocation U(I), introduced in this manner, is not a local property of the point in question onthe dislocation loop. It depends on the dimensions and on the shape of the entire loop. In order to estimate the order of magnitude of U(I), we assume that the function y(~) has a constant value inside a tube of small radius ro, described around the

A. M. Kosevich

94

Ch.l

dislocation line, and equals zero outside this tube. Then there follows from eqs. (149) and (150) the estimate U(l) '"

Jib2

R

4n

ro

-log~,

The dimensionality of the quantity m* is mass per unit length, therefore we can call it the "rest mass" per unit length of a dislocation line. The equations (149) and (150) provide a procedure for calculating the self-energy of a dislocation loop of arbitrary shape. But we shall calculate only the self-energy of dislocation lines of the simplest forms. If we have a straight edge dislocation, then tt' = 1, bt = 0 , bn = 0, and the formula for U(l) may be reduced to eq. (35). b If we 2 . have a straight screw dislocation, then tt' = 1, bt = ± 1, (bn) = 1, and we 0 tam eq. (32). . . Consider a circular prismatic dislocation loop of radius R WIth Burgers vector normal to its plane. Then bt = 0, bn = 0, and the self-energy per unit length, obtained with logarithmic accuracy*, is equal [46J to

c

4n(1 - v)

R

log-· ro

where v = vCr, t) is the velocity of displacement of an element with a coordinate rat the instant of time t. If the dislocations move and the dislocation density varies with time then eq. (154) is not compatible with eq. (116) and we shall therefore replace it with

(155) in which the tensor Jik must be chosen so as to make eqs. (116) and (155) compatible. The condition for the compatibility of eqs. (116) and (155) takes the form of the equation

(156) It is easy to verify that eq. (156) is the differential form of the law of conservation of the Burgers vector in the medium. Indeed, let us consider some stationary closed line s in the medium. We take an arbitrary surface bounded by the line s, and introduce in the formula of eq. (115) the total Burgers vector b of the dislocations that are "coupled" with the surface, that is, enclosed by the line s. Then, we can obtain from eq. (156) in elementary fashion

(153)

The energy of eq. (153) certainly coincides with the energy per unit length of the . straight dislocation, the length of which is of order of magnitude R. The interesting formula for the self-energy of a helical dislocation was obtamed by de Wit [45].

7. Dynamics of a crystal with dislocations 7.1. The dislocation flux density tensor Equation (116), which is the fundamental relation that introduces dislocations into elasticity theory, does not depend on whether the dislocatio~s are s~at~onary or moving. However, it is obvious that in the dynamical case the tune vanat~on of the distortion tensor should be determined essentially by the character of monon of the dislocations.

* The log term

If the dislocations remain stationary when the elements of the medium are displaced we have the obvious equality

(154)

(152)

pb 2 e2

95

(151)

where R is a characteristic radius of curvature of the dislocation line at the point under consideration (in the case of a straight dislocation R; is either its length or the dimension of the body). . . We rewrite eq. (151) in a rather different form using the well-known relatlO~shiP J1 = pc", where p is the density of the medium and e is the transverse velocity of sound. Then we obtain

U =

Crystal dislocations and the theory ofelasticity

in the expression for the self-energy predominates when the radius R becomes large.

(157) From the meaning of eq. (157) it follows that the integral in the right side of eq. (157) determines the magnitude of the Burgers vector "flowing" per unit time through the contour S, that is, carried away by the dislocations that cross the line s. Therefore the tensor Jik can naturally be called the dislocation flux density [47] and eq. (156) is the equation for the continuity of the dislocation flux. Equation (156) was obtained independently by Kosevich [47] and Mura [48J. An equation coinciding in form with eq. (156) was derived by Hollander [49], but it expresses the relationship between somewhat different quantities. The definition of the tensor Jik becomes unambiguous if we note that the dislocation flux density determines directly the rate of plastic deformation of the medium. To verify this, we note that the vector v is the velocity of the total geometric displacement of an element of the medium and determines the rate of the total geometrical distortion uJ: (158)

A. M. Kosevich

96

Ch. 1

With the aid of eq. (158) we rewrite eq. (155) in the form

Crystal dislocations and the theory of elasticity

§7.l

97

It follows from eq. (163) that in the case of parallel dislocations with identical velocities at the point of space in question we have

a T' (U ik - U ik) = -lik'

at

(164)

The difference u~ - U ik determines that part of the total distortion not connected with the elastic stresses, usually called the plastic distortion of the body. Denoting this quantity by Ufk we obtain (159)

Equations (164) and (162) are equivalent. In the case of a continuous distribution of dislocations, the tensor Jik is a continuous function of the coordinates satisfying the condition of eq. (156). The tensor i-; is significant in itself and is a fundamental characteristic of the dislocation motion.

Thus, the change in the tensor of plastic distortion at a certain point of the medium after a short time bt is equal to dl

(160)

~,,---------v

If we write a relation similar to eq. (160) for the plastic strain tensor efk' its form will be

s

(161) A relation equivalent to eqs. (159)-(161), but in a different notation, was indicated by Kroner and Rieder [50]. Let us compare eq. (161) with the expression for the plastic strain tensor eq. (49) caused by a single dislocation and substitute bx = V ot. It is clear that for an isolated dislocation loop the tensor Jik has the form (162)

We may verify the formula of eq. (162) by starting from the physical meaning of the dislocation flux density. We consider a single dislocation with a Burgers vector b, each point of the loop of which moves with a velocity V = VCr), and we calculate the Burgers-vector flux produced by such dislocations when some line s is crossed (fig. 44). If dJ is an element of arc of the line s, and t is the unit vector tangent to the dislocation loop in the vicinity of the point where the line s is crossed, then crossing of the line s by the dislocation loop with transport of the Burgers vector will occur only when there is a velocity component V perpendicular to both dl and t. It is obvious that the number of such crossings of the element dl by the "parallel" dislocation loops per unit time is given by the quantity N(dl x t)V,

where N is the number of dislocations passing per unit area of the plane perpendicular to t. Therefore, the flux of the Burgers-vector component b, through the lines is equal to

1 1 iik

dli =

N(dl

X

t)bk ·

A

Fig. 44 A dislocation AB with tangent vector t moves with speed JIand crosses the element dl of line s.

The average dislocation flux density can be expressed in terms of the same scalar distribution function pP(t) which enters eq. (117):

iik)

= 8 ilm

ff

l,bfv!(t; r)pP(t; r) dO,

(165)

where VP(t) is the average velocity of the dislocation-length elements having corresponding directions of band t, and dO is the differential of the solid angle. Special interest is attached to the connection between the trace of the tensor Jik(JO = Ad and the continuity equation of a continuous medium [47]. The trace j., enters the following equation derivable from eq. (155): div v - (aekklat) = - i;

(166)

It is easy to explain the physical meaning of eq. (166). Indeed, the trace ekk is the relative elastic change in the volume of the medium element, connected in obvious fashion with the corresponding relative change in its density

ekk = -bplp,

(167)

p is the density of the medium. Substituting eq. (167) into eq. (166) and using the linearity of the theory, we arrive at the relation

(163)

aplat

+ div pv = - pJo.

(168)

98

A. M. Kosevich

Ch.l

If the elements of the medium are displaced during the motion of the dislocations without loss of continuity then, by virtue of the continuity equation, the left side of eq. (168) vanishes and ia == ikk

= O.

7.2. The elastic field of moving dislocations Let us set up a complete system of differential equations describing the dynamics of an elastic body in which there are moving dislocations. We assume that the displacement of the dislocation is not accompanied by transport of mass, and take account of the fact that no additional distribution of the concentrated body forces is associated with the dislocation line. Then the equation of motion of the elastic medium can be written as (170) where the elastic stress tensor Pik is connected by Hooke's law eq. (6) with the elastic strain tensor. In eq. (170) we wrote the material time derivative _ aV

-a + (vV)v i = -ati +

ao, Vk - ·

aXk

(171)

Let us substitute eq. (155) into eq. (171) and take into account the fact that the ordinary elasticity theory is linear with respect to elastic strains and rates of displacements. Therefore, we can write d».1 avo dt ~ atl - vkAi'

99

Thus, if the dislocation fluxes are specified, the equation of motion of the elastic medium in the linear approximation has the form (172)

(169)

Comparing eq. (169) with eq. (161), we see that the statement that z, vanishes is equivalent to the statement that the corresponding plastic deformation is not connected with a change in the volume of the body. For linear dislocations, the condition of eq. (169) has a simple meaning. Indeed, in the case of a single dislocation the trace ia is proportional to (b x t) V, that is, it is proportional to the projection of the dislocation velocity on the direction perpendicular to the vectors t and b, in other words, on the direction perpendicular to the glide plane of the dislocation. Thus, eq. (169) signifies that when the medium remains continuous the dislocation velocity vector V always lies in its glide plane and consequently, mechanical motion of the dislocation can occur only in this plane (see sect. 4.). On the other hand, if the dislocation motion is accompanied by formation of certain discontinuities, for example, macroscopic accumulation of vacancies along a section of the dislocation line, then the left side of eq. (168) differs from zero and is equal to the rate of relative inelastic increase in the mass of a certain volume element of the medium (or, accordingly, a decrease in its specific volume).

de, oVi -d = t t

Crystal dislocations and the theory of elasticity

§7.2

In the case of a single moving dislocation the tensor iik is replaced by eq. (162) and in the right part of eq. (172) there appears the term which is proportional to the product vV. Therefore, if terms proportional to v V are taken into account in studies of the dislocation dynamics, then in the right part of the equation of motion (172) the second component should necessarily remain. However, this component is generally omitted and the complete system of dynamical equations of the elasticity theory may be presented in the form: apik aVk - = P -, Pik = CiklmUlm, (173) aXi at (174) If the tensors lX ik and iik are specified, i.e. for specified dislocation densities and dislocation fluxes, the system of eqs. (173)-(174) is complete. The conditions of compatibility of this system are the "conservation laws", eqs. (118) and (156): alX ik -a t

aimk

+ Cilm-a = O. X,

(175)

The system of eqs. (173)-(174) enables us to find Uik (or Pik) and v from any known distribution of dislocations and their fluxes. A system of equations similar to 'eqs, (173, 174) but still different from that given above, was proposed by Hollander [49]. The present system of kinematic equations (174), (175), (158) and (159) coincides with that given in the survey [41J and was employed in the linear theory of dislocation dynamics developed by Kosevich [47, 42J and Mura [48J. The geometrical theory valid for large deformations was developed by Amari [51J. One of the possible ways of solving the system of eqs. (173) and (174) was derived by Nabarro [8J. This method reduces to replacing the infinitely small displacement of an element of the dislocation line by the formation of an infinitely small dislocation loop, and it is convenient when only the time dependence of the deformation field is considered. Perform the differentiation of eq. (173) with respect to time and use eq. (155): (176) We have obtained the dynamical elasticity-theory equation for the determination of the vector v. The vector Pi = Cik1m ailm/aXk in this equation plays the role of the

100

A. M. Koseuich

Ch.l

density of the body forces. The solution of eq. (176) can be represented in the form v,(r, t) =

f dr' fro '§ik(r -

r', t - t')Pk(r', t') dt'

(177)

where ~ik(r, t) is the Green's tensor of the dynamical elasticity-theory equation. Equation (177) completely solves the problem of finding the displacement rates and determines the time dependence of the displacement field. Finally, let us consider the case where dislocation loops are distributed in the crystal in such a way that their total Burgers vector is zero, and the dislocation distribution is described by a dislocation polarization tensor P ik . The dislocation flux density is given in terms of the same tensor P i k by [52, 42] Jik

=

(178)

-OPik/ot.

This is easily seen, for example, by calculating the integral SJik dr over an arbitrary part of the volume of the body, using eq. (162), to give a sum over all dislocation loops within that volume:

fii' dr

= -

:t fPi.

dr.

Comparing eq. (178) with eq. (159), we see, that

c5ufk =

sr.;

(179)

Consequently, in the absence of a resultant Burgers vector in the body, the change in the plastic distortion tensor at each point of the medium is equal to the change in the dislocation polarization tensor at the same point. It must be recalled, however, that there is a fundamental difference between the tensors P i k and Ufk' Whereas the tensor P ik is a function of the state of the body, the plastic distortion tensor is not a function of the state of the body, but depends on the process which has brought the body to the given state. If we agree to regard plastic deformation as being absent in the state with P i k = 0, then, as already mentioned in sect. 6.2., we obtain OUT

U ik

= ~ uX i

P ik,

101

Thus, the question of finding the elastic field produced by dislocations with a zero total Burgers vector reduces to a known problem of elasticity theory. When the plastic polarization tensor P ik is specified, the vector u T can be determined from the dynamical elasticity equation expressed in terms of displacements, in which the density of the body forces is J; = - Cikl l1l oP II1l/OXk. The solution of eq. (182) can be written similarly to eq. (177), namely

u,T() r, t = ckl mn

fd 'ft r

,£! ;';1ik( I'

-

I 1',

er; d'

t - t -0- t. ')

(183)

Xl

-00

Equation (183) in conjunction with eqs. (180) and (181) completely solves the problem of finding dynamical displacement and strain fields, and the rates ofdisplacement.

7.3. The stress field in the linear approximation of the dislocation velocity Assume that the dislocations occupy a certain part of the elastic medium, moving with small velocities V in a region with linear dimensions L. We shall assume that V « c, where C is the speed of sound in the medium, or, to put it another way, L « A, where A is the characteristic length of the sound wave generated by the motion of the dislocations, and expand all the quantities of interest to us in powers of the delay time of the sound waves inside the dislocation system, confining ourselves to the first terms of the expansion. Such an expansion is valid both inside the dislocation system and at distances R ::::: L from the system. We write down the stress tensor in the form Pik = p~~) + P~t), where p)f) denotes the quasistatic stress field obtained when the delay is completely neglected, and p~i) is proportional to the time derivatives of the dislocation polarization tensor and describes the dynamical effects. We shall not write out in an explicit form the quasistatic part of the stress tensor, since it is treated widely in the literature [6, 43]. We note, however, that this part of the stress tensor can be represented in the form

(180)

where u T is again the vector of the total geometrical displacement from the position in the undeformed state. The rate of displacement v is in this case connected in the usual fashion with the time derivative of the vector u T :

v = OUT/ot.

Crystal dislocations and the theory of elasticity

§7.2

(181)

We substitute eq. (180) and (181) into the equation of motion (173) and use Hooke's law: (182)

R

= II' - 1"1,

n = (I' - r')/R,

(184)

where the fully defined tensor function qiklm('n) contains only the components of the vector n and dimensionless factors of the order of unity. As to the dynamical part of the tensor P ik» we represent it in a shortened form as [53] (1) _

Pik

-

p 0 4n ot

f

R (). (') fJiklm n IJl m r

dr' R'

(185)

A. M. Kosevich

102

Ch.l

where the tensor fJiklm(n) in an isotropic medium is connected with the components of the vector n by the relation fJiklm(n) = t[(1 - 2y2

+

Z

+

in, bk1 + nk bil)nmJ

1=

y4)(b il bkm + bim bk1)·

Here y2 = c; Ic;; Ct is the transverse velocity and c, is the longitudinal velocity of sound. For simplicity we have confined ourselves to the case Ak = 0, assuming that the dislocations move in their slip planes. In an anisotropic medium eq. (185), naturally, remains valid and the connection between the tensor fJiklm(n) and the elastic moduli of the medium has been determined for the general case by Kosevich and Natsik [54].

103

energy radiated by the system of moving dislocations in an isotropic medium is equal to [47J

3y4)bik - 3(1 - y4)ninkJnznm

- t y4[(ni bkm + nk bim)n + i(1 +

Crystal dislocations and the theory of elasticity

§7.4

5~C

HGJ [jj~I

+ [jj~J}

(187)

The bar in eq. (187) denotes averaging over a region of space whose dimensions exceed the radiation wave-length. The radiation of elastic waves by a system of moving dislocations in an anisotropic medium can also be considered in a general form [52]. The radiation by a single straight dislocation can be obtained more simply by using other methods. Eshelby [55, 56J has considered the radiation from a screw dislocation. Formulae for the radiation from a straight edge dislocation can be derived from the results by Kiusalaas and Mura [57J*.

7.4. The radiation field of moving dislocations Let us consider the deformation produced in an unbounded isotropic medium by a system of moving dislocation loops at large distances R considerably in excess of the dimensions L of the system. We assume that the speed V of the dislocations is small compared with the velocity of sound C in the solid. Then the ratios LIR and Vic are small parameters, characterizing the deformation field. We confine ourselves to the first nonvanishing terms of such an expansion, assuming the smallness of both parameters to be of the same order of magnitude. We choose the origin somewhere inside the system of dislocations and denote by R o the distance from the origin to the point of observation (no is the unit vector in the appropriate direction). Then by virtue of the smallness of the dimensions of the system, we can put R ~ R o - r'n., in expressions similar to eqs. (177) and (183) and expand the integrand in powers of r'n. We confine ourselves to the case Ak = O. Then, at distances R that are large compared with the wavelength of the emitted elastic waves (R o » cLIV), the displacement vector in the radiated sound wave has the form [47J 1 u = -2-ncRo

{(C-

t)3.

n[nDJt

Ce

+

.

.}

[D - n(nD)Jt '

(186)

where the dot denotes differentiation with respect to time. The vector D in eq. (186) denotes the "projection" of the symmetrical part of the total dislocation moment tensor D ik (see sect. 6.2.) on the direction of n: D,

= nkD~i'

D~k

= i(D ik + DkJ·

The square brackets with the subscript t in eq. (186) enclose the quantities taken at the instant of time t - Rolcp where Ct = C is the transverse velocity of sound. The square brackets with the subscript t in eq. (186) contain quantities taken at the instant of time t - Rolce, where Ce is the longitudinal velocity of sound. With the aid of eq. (186), using the well-known formula for the flux density of sound energy [10J, we calculate the intensity of radiation. The total elastic-wave

8. Equation of motion ofa dislocation 8.1. The field nature of the equation of motion of a dislocation The system of eqs. (173)-(174) formulated in sect. 7.2. determines the elastic distortion tensor Uik and the material displacement velocity vector v from the known distribution of the dislocations and their fluxes. The tensors (Xik and Jik' describing the motion of the dislocations, are assumed specified. In order for the system of eqs. (173)-(174) to be completely closed and descriptive of the self-consistent evolution of the dislocations and of the elastic field, it is necessary to show how the densities ofthe dislocations and of their fluxes vary under the influence of elastic fields. In other words, it is necessary to derive an equation of motion for the dislocations. By virtue of the definition given above for a dislocation, the equation of motion of a dislocation should be derivable from a field. The field-theory aspects of equations of motion and of the corresponding masses are universally known and will not be developed here. There are many ways of deriving the equation of motion of a dislocation. In the survey [42J the author discussed in detail the derivation of this equation by analogy with the Lorentz derivation of the equation of motion of the electron, i.e. starting from the notion of self-action of the dislocation. We present only the final result of the derivation [53]. Let us consider a dislocation element, which has a tangent vector t and moves with a certain velocity V. The motion of this element takes place subject to the condition that CikltkPlmbm

+

pvb(t x V)i

= 0,

(188)

where all the quantities are taken at the point at which the dislocation element is situated. * See Addendum 7.4.

104

A. M. Kosevich

Ch.l

The tensor Pik and the vector v contained in eq. (188) include both the external elastic field and the self-field of the dislocation loop. Thus, eq. (188) relates the dislocation self-field, meaning also the dislocation-loop motion that generates this field, to the external fields. Inasmuch as the external fields in eq. (188) are taken at the same point of space, at which the dislocation element under consideration is situated at the given instant of time, eq. (188) is an implicit equation of motion of the dislocation. The expression (189) which when set equal to zero constitutes the equation of motion (188), determines the total force exerted on' a unit length of the dislocation. The same equation was obtained by Rogula (58] in the model called the model of a "pseudo-contiauumvby.using a different technique. In order for the equation of motion (188) to have the usual explicit form, that is, for it to relate the instantaneous coordinates and velocity of the dislocation with the instantaneous value of the external stress field, it is necessary to express the selffield of the moving dislocation at some instant of time in terms of the coordinates and velocity of the elements ofits loop at the same instant of time. We know that this can always be done it) the-approximation that is linear in the dislocation velocity, if the set ofequations (173) alfd(l74) is used. In the discussion ofthe set of equations (173)-(174) it was noted that eq. (173) can be obtained from eq, (170) if the terms proportional to vV are omitted. Therefore, following this approximation, we should omit the second term in the right-hand side of eq. (189) and write : (190) Details concerning the form of the expression giving the force of elastic origin which acts on the moving dislocation have been discussed by many authors. An analysis of this discussion is given in the paper by Malen [59]. Before discussing the explicit form of the equation of motion and introducing the effective mass ofthe dislocation it should be noted that the equations of motion (188) or (190) involve only nondissipative forces of elastic origin. As to the motion of a real dislocation, it should be said that this dislocation experiences also some forces of inelastic origin.

Crystal dislocations and the theory ofelasticity

§8.2

105

dislocation stresses. We transform the self-force F(flS) in eq. (190), assuming the dislocation line to be not infinitesimally thin, but of some finite thickness, with a Burgers vector which is "smeared" over this thickness*. Then by using eq. (146) we can write the force of self-action F(pS) in the form:

fP~mY(~) d2~.

F,U)') = B'kltkbm

(191)

We now use for the tensor P~k an expansion in powers of V and represent it as a sum of two terms: P~k = p~Z) + p~~), the first of which p~Z) determines the quasistatic stress field, and the second one P~t) the stresses proportional to the acceleration of the dislocation. The term p~2) characterizes the linear self-tension of the static dislocation loop. This force can be easily calculated by substituting in eq. (191) the static expressions for p~Z). Since this operation is perfectly obvious, and since the quasistatic tension force of the dislocation loop has itself no bearing on the question of interest to us, that of the effective mass of the dislocation, we shall not write down an explicit expression for this force. We note, however, that the self-tension force corresponds to a definite self-energy of the dislocation at rest. The second part of the self-action force is determined by the term p~~) and can be expressed with the aid of eq. (185) in terms of the dislocation acceleration. Before we write this expression, we shall make two remarks. Firstly, the mechanical motion of the dislocation along the glide surface presents the greatest interest since this motion can occur with large velocities and accelerations. The displacement of the dislocation along the normal to the slip plane can go quickly only in the case of the dynamical development of a crack. However in the latter case the continuity of the medium is broken. We shall not discuss such problems here, and shall confine our consideration to dislocation glide. Secondly, in writing down the dynamic self-action force of the linear dislocation we shall be interested only in the terms having a logarithmic singularity. With such considerations it is possible to separate the main part of the force F(pS) within a logarithmic accuracy, which is quite sufficient for us. Then, substituting in eq. (191) the indicated part of P~t), we can write for the corresponding part of the self-action force F/jj(l»)

~

f

1',,(/, I') Wi I') dt',

-

W(l) = V(l),

(192)

where W(l) is the acceleration of the dislocation element and 8.2. The explicit form of the equation of motion of a dislocation

In order to have the equation of motion of the dislocation in the usual form, it is necessary to separate explicitly in eq. (188) the force of self-action of the dislocation, and to exclude the singularities of the dislocation self-field, in a similar way to that which is done ill the derivation of the field mass and the equation of motion of an electron by the Lorentz method. To take the self-action into account, we represent the field Pik in eq. (190) in the form Pik = P~k + P~k' where P~k is the external field, and P~k is the self-field of the

1',,(/, I') b cos

~ tPb (tf li" 2

e = nb,

1 r(l,I') = 4n

ff

n

- t;t,{

1+ (~T 0] tu, sin?

= [x(l) - x(l')]jR(l, 2

I'),

t

1'),

= t(l), t' = t(l'),

2

;: d (' y(~)y(~') dReI, I') .

(193)

* It is obvious that, as in the case of static dislocations, the self-action of moving strictly linear dislocation; (that is dislocations with zero thickness) is described by divergent integrals.

Ch.1

A. M. Kosevich

106

We note that in eq. (193) as.in eq. (149) the vector n is taken outside the sign of the integration with respect to ~ and ~/. The expression for flik(l, 1/) in an anisotropic medium was obtained by Kosevich and Natsik in [54]. Substituting eq. (192) into eq. (190) and taking into account.the.presence of forces of inelastic origin, which were mentioned above, and which also act on the dislocation, we write down the final form of the equation of motion of the element of the dislocation loop:

t

Jlik(l, I') Wk(l') dl'

~ F,o(l)

+ "'km t k(l)P~n(l)bn +

S ,(l, V)

(194)

where Fa is the already-mentionedquasistatic dislocation self-tension, and S is the force of inelastic origin, which, naturally, depends on the dislocation velocity. It is important to note that the inertial term in the equation of motion, that is, the left side of eq. (194), plays an important role only in the case of sharply nonstationary motions of the dislocation, when the acceleration of the dislocation is very large. If the acceleration of the dislocation is small, then the major role is played by deceleration forces (resistance forces), which include the dissipative forces. It is the magnitude and the dependence on the dislocation velocity of these forces which determine essentially the character of the almost-stationary motion of the dislocation. We now consider explicitly a dislocation which moves along its glide surface. We introduce at each point of the dislocation line a right-hand triplet of unit vectors (t, 'V, K), v being the vector normal to the glide plane and K the vector normal to the dislocation line in the glide plane. A contribution to the self-action force is made only by the dislocation velocity (or acceleration) component perpendicular to the dislocation line. In the case in question, these components are equal to

Therefore, multiplying eq. (194) by tJl(I, I')W,(I') dl'

K,

we obtain

~ Fo(l) + vm(l)p~"(l)bn + S(l, V,),

(195)

=

iPb2(tf')(KK'{ 1

+

GJ

sin?

J

e

T(l,l').

(196)

An expression in lieu of eq. (196) for fl(l, 1/)in an anisotropic medium was obtained in [54]. Equation (195) describes the mechanical motion of a dislocation loop in an external stress field. In the derivation of the equations of motion of the dislocation we used eq. (185), which is the second term in the expansion of the dynamic dislocation field in the delay of the elastic waves, i.e. formally in powers of lie. Consideration of the next terms of this expansion allowsus to calculate the retarding force due to radiation of sound

107

waves by a nonuniformly moving dislocation. Natsik [60J has shown that this force is similar to the corresponding radiation force for a nonuniformly moving electric charge and is proportional to the time derivative of the dislocation acceleration.

8.3. The effective mass of a dislocation can conclude from eq. (194) that flik(l, 1/) has the meaning of the nonlocal effective field mass density of a dislocation line. The physical meaning of the tensor function flik(l, s) is that it establishes the contribution made by the kth component of the velocity of the element with coordinate s on the dislocation loop to the ith component of the momentum of an element of unit length with coordinate I on the same loop. The physical meaning of the quantity flik(l, 1/) can be described from a somewhat different point of view by writing down the kinetic energy of the dislocation loop Ek'n

=

(197)

i f f Jlik(l, I') V,(I) ViI') dl dl'.

From eq. (197) the obvious symmetry of the tensor flik(l, s) follows which, naturally, is implied in its definition in eq. (193): (198) If we compare flik(l, 1') with the expression for the density of the dislocation selfenergy eq. (149), we can easily verify that flik(l, 1') and 8(1, 1') have a similar functional dependence on position on the dislocation loop, and that in order of magnitude fl(l, 1')e2 '" 8(l, 1/).

The concrete expressions, however, are different and therefore the dynamical nonlocal mass density does not reduce in the general case to a nonlocal "rest mass" density. Only for a straight screw dislocation do we have 2 flik(l, 1')e

where for an isotropic medium we get with logarithmic accuracy Jl(l, I')

Crystal dislocations and the theory of elasticity

§8.2

= bik8(l, I').

Along with the equation of motion of the dislocation element eq. (194), we can consider the averaged equation of motion of the entire dislocation loop by integrating eq. (194) over the loop tm'k(l)Wk(l) dl = F,',

F,' = "ikmbn

ttkP~n dl +

ts,

dl

(199)

where Fe is the total external force acting on the dislocation, and m'k(l) = tJl'k(l, I') dl'.

(200)

The first term in the right side of eq. (194) drops out after integration over the entire loop, since the total self-action force of a dislocation at rest is equal to zero. If the applied stress is uniform, there is also no resultant elastic force on a closed dislocation loop. If the stress field varies only slightly over the loop, the total elastic

A. M. Koseoich

108

Ch.l

force acting on the dislocation and entering into eq. (199) is determined by the elastic stress gradient [14]. It follows from eq. (199) that when the motion of the entire dislocation loop is considered, mik(l) plays the role of an effective mass per unit length of the dislocation. It is obvious, however, that the effective mass per unit length of the dislocation, introduced in this manner, is not a local property of the point in question on the dislocation loop. It depends on the dimensions and on the shape of the entire loop. Using eq. (200) and the definition of the tensor /lik in eq. (193), let us estimate the order of magnitude of m ik. For such an estimate we shall assume that the function y(,) takes a constant value inside a tube of a small radius r0' including the dislocation line, and is equal to zero outside this tube. We can then readily obtain the estimate m '"

pb 2

R

4n

ro

-log~,

(201)

where R, is the characteristic radius of curvature of the dislocation line at the point under consideration. In the case of the rigid motion of a straight-line dislocation, R, is the dislocation length. If the dislocation oscillates, then R, is equal to the wavelength of the bending oscillations of the dislocation line [61]. In order of magnitude m ik '" m*, where m* is the rest mass per unit length of the dislocation defined by eq. (152). The estimate obtained for m ik justifies our assumption of a pure field mass for the dislocation. The point is that when a real dislocation moves in a crystal, it sets in motion also some of the atoms in the vicinity of the dislocation axis, at a distance of the order of r0 from the axis. This produces an additional dislocation inertia, connected with the ordinary mass of these atoms. The order of magnitude of the mass of the atoms inside a tube of radius r0 '" b can be estimated as pr6 '" pb 2 per unit length of the dislocation. Comparing this estimate with eq. (201) we see that when log (Relr 0) » 1, taking into account the masses of the moving atoms near the dislocation line does not noticeably change the dislocation inertia, and the dislocation mass can be actually regarded with logarithmic accuracy as a field mass. For a straight-line dislocation, when the vectors t and b are constant along the dislocation line, the tensor m.; is described by the very simple expression

(202) where R m denotes the length of the dislocation. In the case of a screw dislocation, eq. (202) leads to the expression previously derived for the screw-dislocation mass by Frank [62J and Eshelby [56]. Finally, let us consider the motion of a dislocation loop as a unit, (203)

Crystal dislocations and the theory of elasticity

§8.3

109

by introducing the average acceleration W o of the dislocation loop and the tensor of its total mass M ik , defined by the formulae

MikW~ =

fmik(l) Wk(l) dl,

u.; ~

fmik(l) dl.

(204)

The equations of motion (194) and (203) enable us to separate the study of the motion of the dislocation loop as a whole from the study of the relative motion of its elements. Having derived the equation of motion of the individual dislocation, we have, in principle, a complete system of equations defining the evolution of a set of dislocation loops and the elastic field in the solid. However, the dislocation equations of motion in this system are equations of motion of the discrete structures. Yet the entire theory of the dynamic elastic field produced by the dislocations has been formulated in terms of a continuous dislocation distribution. A natural way of recasting this system of equations in a unified form is to average the equations of motion of a large number of individual dislocations and to transform them into equations of motion of continuously distributed dislocations. This problem was discussed in the survey [42]. 8.4. Dislocation damping in a medium having dispersion of the elastic moduli

Dislocations moving in a crystal always experience a damping force, leading to energy loss. It is necessary to take this force into consideration when we write the equation of motion of the dislocation. There are many mechanisms of dissipation which retard the motion of a dislocation. Some of these relate to the discrete structure of the crystal and the detailed structure of the dislocation core. However, there exists a damping mechanism which can be analyzed by considering the crystal as a continuous medium. It relates to the mechanism whereby dislocation energy is dissipated due to relaxation processes occurring in the deformation field of the moving dislocation. The corresponding energy loss should be considered within the framework of the continuum theory of moving dislocations. Unfortunately, there is at the present time no systematic theory for the dislocation damping produced by the discrete structure of the crystal, so that it is not possible to determine in all cases the relative importance of the contribution from relaxation mechanisms to the dislocation damping force. We can only expect that the dislocation energy loss by relaxation processes will be quite substantial for a dislocation velocity V lying in the range c(a s / /l) 1/2 « V « c, where as is the "starting stress" for initiation of dislocation motion, /l is the shear modulus of the crystal (/l » as always), and c is the velocity of sound. The lower limit of this range is determined by the following condition: in the range of dislocation velocities of interest here, the kinetic energy of the dislocation should substantially exceed the rise in the interatomic interaction potential, which determines the starting stress. Regarding the high velocity limit, it is connected in a natural way with the fact that at V '" c the motion of a dislocation should be described by theories of a discrete crystal lattice, and not by a macroscopic theory. In other words, at such velocities the dislocation retardation should be essentially determined by that component of the dislocation damping which is

110

A. M. Koseoich

Ch.1

directly connected with the atomic structure of the material and not with its average properties. In the indicated dislocation velocity range, where the relaxation energy loss can be substantial, it makes sense to analyze it separately. The justification for such a separation of the macroscopic damping force is as follows. In the first place, each relaxation mechanism is characterized by its own specific relaxation time r, which depends on the state of the crystal (in particular, on its temperature), and, generally speaking, is unrelated to the characteristic lattice frequencies: therefore, with dislocation velocities of order V "" ali, where a is the lattice constant, we would expect an increase in the damping force due to these relaxation mechanisms. The -position and size of the corresponding maximum or region of nonmonotonic behaviour in the velocity dependence of the damping force is determined by the state of the crystal (for example, its temperature), and consequently should change in a controllable way. In addition, it is possible to have relaxational energy loss connected with contamination of the crystal, i.e. with the presence of impurities. The contribution of these losses to the damping force, on the one hand, depends on the impurity concentration, and therefore in principle can be separated from the others. On the other hand, such losses sometimes can have a peculiar, almost resonant, nature, if the impurities have quasidiscrete eigenfrequencies, and this should generate specific features in the dependence of the damping force on the dislocation velocity [63]. Additional damping mechanisms due to relaxation phenomena in the elastic field of a moving dislocation have been analyzed previously by various authors. Eshelby studied thermoelastic absorption of energy related to the thermal conductivity of the medium (thermoelastic friction) during the vibrational motion of edge dislocations [55J, and also found the damping force on screw dislocations moving in an isotropic viscoelastic material (the so-called "standard linear medium") [64]. Schoeck and Seeger have considered [65J the dissipation of energy due to local changes in the order near dislocations in a solid solution. Mason [66J has estimated the effect on dislocation damping of the exchange of energy between the various branches of the phonon spectrum, which come into equilibrium with each other during the deformation (phonon viscosity)". Here we present a theory of relaxation damping which does not particularize the relaxation mechanisms and is valid for crystals of any symmetry. The method of calculation used for the damping forces is similar to the electrodynamic calculation of the ionization loss of a charged particle passing through matter. It is known that the presence of relaxation of the elastic stress in a continuous uniform medium leads to dispersion of the elastic moduli [67]. This means that Hooke's law for such a medium in a literal sense can only be written for stresses Pik and strains eik varying harmonically in time, P, e o: exp [ - iwtJ: (205) The components of the elastic modulus tensor Cik1m(W) are, generally speaking, complex functions of the frequency co. Just as the presence of an imaginary part of

* See also Ref. [127].

§8.4

Crystal dislocations and the theory of elasticity

111

the dielectric constant in a medium leads to absorption of energy from electromagnetic waves, the imaginary parts appearing in the components of the tensor Cik1m(W) lead to the absorption of sound vibrations by the medium, i.e. cause their energy to be dissipated. The relationship between the internal friction and the dispersion frequencies of the elastic moduli of isotropic media has been analyzed by Zener [67]. In those cases where it is necessary to take into account spatial dispersion of the elastic moduli also, an equation of the type of eq. (205) should relate a stress and a strain whose dependence on space and time is of the form exp [i(k,. - wt)]. In other words, the elastic moduli should be considered to be functions of the frequency w and the wave number vector k, and Hooke's law should be written in the form Pik

=

Cik1m(W,

k)e 1m ·

This spatial acoustic dispersion should be taken into consideration when subsonic, "transonic" and supersonic motion of the dislocation is analyzed [68, 69J*. It is clear that the damping force on a dislocation in a medium with dispersion of the elastic moduli can be expressed in terms of complex elastic moduli. In order to carry this out it is necessary to calculate the elastic stress created by a moving dislocation in such a medium and to determine the force exerted by this field on the dislocation. This problem is solved by Kosevich and Natsik [70J for a straight dislocation moving with a constant velocity in the medium with an arbitrary symmetry. In an isotropic medium the damping force can evidently be expressed in terms of the complex elastic moduli. We assume that the mechanisms which give rise to dispersion of the elastic moduli have negligible spatial dispersion. Then the dispersion of the moduli leads only to a frequency dependence: A = A(W) and j1 = j1(w). The damping forces on an edge dislocation FE and screw dislocationF, have the forms [70J : F = - ibi J~ (kV) A(kV) + j1(kV) dk E n · Ikl j1 A(kV) + 2j1(kV) ,

ib~

F s = - 2n

J1kfk j1(k V) dk,

(206)

where bland b 3 are the Burgers vectors of the edge and screw dislocations, respectively. We separate the real and imaginary parts in the right-hand sides of eq. (206). From general consideration, it follows [67J that the real parts of the complex elastic moduli should be even, and the imaginary parts odd functions of w; therefore, the real parts of the integrands in eq. (206) are odd, and their integrals over all values of k are zero. The imaginary parts of these integrands are wholly due to dispersion of the elastic moduli and will be even functions of k. Consequently, we can double the integrals and integrate over positive values of k. In speaking of the integrations in eq. (206), which take into account all possible values of k, it is appropriate to estimate the upper limits of these integrals. We know that the following proportionalities usually appear for very high frequencies w:

* See Addendum 8.4.

A. M. Koseuich

112

Ch.l

Im A OC ljw and Im u o: ljw; therefore, eqs. (206) are logarithmically divergent for large k, i.e. for large k V. In order to understand the physical reason for this divergence,

we note that integration over very large k values corresponds to considering the deformation in a very small neighbourhood of the dislocation core, where the lattice distortion cannot in principle be described by elasticity theory. The contribution to the damping force from deformation in the immediate vicinity of the dislocation core should be considered within the framework of microscopic models, which is not our aim. Rogula [69J showed that a reasonable formulation of the model of the elastic continuum having acoustic dispersion eliminates the divergence of integrals of this kind. In order to take into account correctly the range of applicability of the ideas used, and eliminate the non-physical divergence of the integrals, it sufficesto cut off the integration at some wave number k = k o, where k o is of the order of magnitude of the inverse linear dimension of a dislocation core (kob '" koa '" 1). In an isotropic medium it is convenient to transform from complex elastic moduli to quantities having a direct physical meaning, namely, to the attenuation coefficients and phase velocities of sound waves. We consider the case where the absorption has a nonresonant character and it is small in the frequency interval of interest (0, k o V). This dispersion of the phase velocity of sound is also small and we can assume this velocity to be the usual isothermal velocity of sound waves. Then [( 1 - 2c?) F E = - -4pbic; -

c;

nV

2pb~c;

f.kOV --dw+ytCw) Ct f.kOV ylw) ] --dw, co

0

cf

0

co

(207)

f.kOV

ytCw) (208) --dw nV ° co ' where ytCw) and yl w) are the attenuation coefficients per unit distance for a transverse and longitudinal sound wave, respectively*. In a real crystal several different relaxation mechanisms can act simultaneously, leading to dissipation of elastic energy. Each of them gives a certain contribution to the absorption and dispersion of sound, and consequently to the damping force on a dislocation. If we limit ourselves to the consideration of frequency dispersion, then for sufficiently general assumptions each dissipation mechanism can be described by its own relaxation time T j and contributes a term of the following form to the sound absorption coefficient: Fs

= ----

W

y/w) = A j 1

Crystal dislocations and the theory of elasticity

§8.4

113

Using formulas of the type of eq. (209) for the sound absorption coefficient we can carry out a qualitative analysis of the way in which the dislocation damping force depends on the velocity. Since the absorption from relaxation processes has, as a rule, a small value (Ac « 1), the calculation of the damping force can be carried out without considering the dispersion of the phase velocity of sound, i.e. using eqs. (207) and (208). We will consider the relaxation time to be independent of frequency and substitute eq. (209) into eqs. (207) and (208). In the result it turns out that the force of relaxational damping is described by a sum of expressions of the form b2 p C3

F.(V) '" A. --log (l J J TjV

+

k~TJ V 2).

(210)

We will estimate the order of magnitude of the second term in the logarithm in eq. (210). Note that koTV '" V(Tja). For all relaxation processes of a macroscopic nature at velocities V « c the quantity koTV can have, generally speaking, a value of the order of or greater than unity. Thus, even for comparatively low dislocation velocities it is possible to have a nonlinear dependence of the damping force on the velocity V. The nonlineardependence of eq. (210), as can be easily seen, leads to the appearance of a maximum in the relation Fj = F/V) at velocities V '" Ij(koT j). The damping force due to those mechanisms for which the inequality koTV:> 1- is fulfilled simultaneously with the inequality V« c, after attaining a maximum, falls off as IjV. Figure 45 shows the characteristic function which describes the dependence of the damping force on velocity: F(V) = const l/J(koT V), where I/J(x) = (ljx) log (1 + x 2 ) . Note that the nonmonotonic dependence of-therelaxational damping force on the velocity, even in the comparatively low-velocity region, has been emphasized by Schoeck and Seeger [65J, and also by Eshelby [64]. If the dislocation velocity V and the relaxation time Tj are such that kOTjV « 1, then the expression for the corresponding term in the damping force has the-order of magnitude F, '" pc3b2k~AjTjV '" pc 3A jTjV, and the relaxation damping force is proportional to the dislocation velocity. In other words, the damping is described by the usual resistive force F = - (X2 V, where the

2T.

+ W ~ T 2'

(209)

1.0

j

If several relaxation mechanisms act simultaneously, with different relaxation times, the absorption coefficient is described by a sum of terms of the type of eq. (209). The quantity A j which enters in eq. (209) depends on the structure and state of the medium, and also on the form of the relaxation mechanism. In addition, both A j and the relaxation time T j have different values for longitudinal and transverse sound waves.

* If the dispersion law for sound waves is represented in the form coefficient y(w) is defined as the quantity y(w)

=

1m k(w).

0.5

o

5

x

10

k = k( w), then the attenuation

Fig. 45 Graph of the function t/J(x)

= (l/x)

log (l

+

x2 ) .

Ch.1

A. M. Koseoich

114

coefficient ry.2 is naturally proportional to the relaxation time r. However, judging from the dependence shown in fig. 45, we should expect that for low velocities (k o Vr « 1) eq. (210) should be used only for coarse quantitative estimates. Actually, in this case the force F is very sensitive to the choice of the parameter k o , which cannot be precisely determined. Also, with koT: V :> 1 the dependence of the calculated force F on this parameter is only logarithmic, i.e., very weak; therefore, strictly speaking, eq. (210) is valid with logarithmic precision and usable in explaining the quantitative dependence only for dislocation velocities V :> air.

There exist many dynamical problems in dislocation theory which can be analyzed by using the so-called string model. This model treats the dislocation line as a heavy string under tension lying on a "corrugated" surface. The corrugated surface describes the Peierls potential. Troughs of this surface correspond to the potential minima on the glide plane occupied by a straight-line dislocation in equilibrium (fig. 46).

- - - - -I - - - l l = a

- - - - -

Max-------

1]

=

Min~l DIslocatIon x

(a)

(b)

Fig. 46 Two types of the motion of a dislocation in the field of the Peierls potential. (a) The dislocation oscillates in its own trough. (b) The dislocation forms a kink moving along the x-axis.

1]0

w2

=

w6 + c6 k 2 ,

21] 21] 0 0 . ( 1]) m ot 2 - T D ox 2 + bo.; sm 2rc -;; = ba,

(211)

where a p is the Peierls stress and a is the corresponding component of the applied stress. The properties of this equation have been discussed by a number of writers [71J. Equation (211) is a non-linear equation which describes rather well dislocation displacements 1] of the order of magnitude of the lattice period. In the absence of an external field a, the dislocation motion takes the form of free oscillations corresponding to the normal modes of vibration of a stretched string. Since the system is non-linear, these oscillations cannot be directly superposed. The lowest modes of these motions are small-amplitude disturbances, when the dislocation oscillates in one trough of the Peierls potential (fig. 46a). In this case 1] « b and the equation of free oscillations reduces to

0ot1]2 2

m

0ox2 + 2rc (b)-;; a 2

TD

1]

p1]

= O.

(212)

(214)

where (215)

= Znbo pima,

Other simple modes describe motions of the dislocation when one end of the dislocation (x = - (f) is situated in one trough of the Peierls potential (1] = 0) and the other (x = (f) is in the next trough (1] = a). This situation is shown in fig. 46b and corresponds to the motion of a single kink on the dislocation (see sect. 4.3.). It.is evident that a kink travelling along the dislocation is described by a solution of eq. (211) satisfying the conditions (f)

= 0,

(216)

= a.

1](00)

It is interesting to note that the solution of eq. (211) under the conditions ofeq. (216) is equivalent to the problem ofthe motion ofthe dislocation in the Frenkel-Kontorova model [73J. A single kink moving with a constant velocity v along the x-axis initiates transverse displacements of the dislocation 2a

Let the x-axis be directed along the equilibrium position of a straight dislocation, and the transverse displacements of the dislocation 1] go along the y-axis in fig. 46. If the dislocation has a mass m per unit length, line tension T D and is subjected to a Peierls potential (see sect. 5.5.), the equation of motion of that dislocation has a form

(213)

sin (kx - wt),

which have the dispersion law

1]( -

Max

115

Leibfried [72J used eq. (212) in considering the thermal motion of the dislocation. The solution of eq. (212) describes sinusoidal oscillations of the type

w6

8.5. The string model

Min - - -

Crystal dislocations and the theory of elasticity

§8.5

1](x, t) = ~ tan

-1

exp

x - vt

(0 _ v2 /C~)1/2 '

(217)

where Co is determined byeq. (215) and

( = co/WOo

(218)

The quantity ( may be called the half-width of the stationary kink. The total energy of the moving kink depends on its velocity v and is [74J

Wk =

f

oo

-00

[1zp (01])2 1 (01])2 ab ot + zT ox + ~ a D

p

.

2rc1]]

sin -; dx

= Wo/O - v2 /C6)1/2,

(219)

where W o = (2/rc 2 ) (a 2 T D j( ) is the energy of the stationary kink. If v « Co, the kink energy of eq. (219) may be written in the form Wk(v) = W o

+ !M kv2 ,

(220)

where

M; = W o/c6· It is natural to call the quantity M k the mass of the kink [74].

(221)

116

A. M. Koseoich

Ch.l

Thus the kink can be considered as a particle constrained to move along the dislocation and having the mass M k • The kink motion along the dislocation causes a transverse displacement of the dislocation. Therefore the kinetics of a set of kinks on the dislocation determines one of the possible mechanisms of dislocation motion under weak applied stresses. Now consider dislocation vibrations effected by an external oscillating force under conditions where Peierls-Nabarro forces may be neglected* and where allowance for the damping force proportional to the dislocation velocity becomes essential. In this case the motion of a dislocation is governed by an equation similar to eq. (211) in which the term coming from the Peierls potential is omitted and the damping term B(a1]/ot) is added, where B is a certain factor dependent on the nature of the dissipative forces. As a result we have the equation m

0

21]

ot

0

2 -

21]

To ax 2

+

B

01]

at = ba., e

- icot

.

(222)

Suppose that the dislocation is pinned by point impurities at x = ±il. Then the solution for a forced vibration of the dislocation segment is 1]

= b(Jo e- icot (cos kx _ 1)' k 2 To

cos ikl

(223)

117

Crystal dislocations and the theory of elasticity

§9.1

Consider for definiteness a simple cubic lattice of a dielectric or metal crystal. An interstitial atom in that lattice leads to such a local distortion of the perfect crystal that all atoms nearest to the interstitial atom are displaced outwards from the interstitial atom (fig. 47a). In a simple cubic lattice this deformation has a cubic symmetry due to the fact that the atoms in the nearest neighbourhood of the interstitial atom experience the action of very symmetrical repulsive forces. The system of these forces has, naturally, a zero resultant and a zero total twist moment. From the macroscopic point of view their action is equivalent to the action of three equal double forces without moment located in the position of the interstitial atom and directed along the coordinate axes (fig. 47b). In elasticity theory this system would be described by a density of body forces in the following form (the defect being placed at a point r = r o) [10, 40J: fer)

=-

KD o grad

oCr - r o ) ,

(224)

where K is the bulk modulus and the quantity Dois equal to the increase in the crystal volume due to the presence of a single interstitial atom in the crystal. An additional atom may only increase the crystal volume and hence, Do > O. The volume change produced by the interstitial atom is usually of the order of magnitude of the atomic volume, therefore, [20 "-' a 3 •

where k 2 = (mw 2

+ iBw)/To·

Equation (222) and its solution eq. (223) are the basis of the dislocation theory of internal friction suggested by Koehler [75] and extended later by Granato and Lucke [76J.

(a)

Fig.47

9. Dislocations and point defects

(b)

A model of an interstitial atom. (a) Atom displacements near the inclusion in a simple cubic crystal. (b) The equal double forces without moment.

9.1. Elastic interaction of dislocations with point defects

A point defect is usually understood as any distortion or imperfection of the crystal lattice concentrated in a volume of the order of magnitude of the atomic volume. However, our present analysis is limited to the most trivial types of point defects in the crystal without impurities. Such are interstitial atoms and vacancies. For most problems dealing with the macroscopic mechanical properties of solids the consideration of static elastic deformations caused by the point defect at distant points is decisive. Generally speaking, the field of atomic displacements around the defect is determined by the nature of the interaction ofthe defect with the surrounding lattice, but, as will be shown below, it may be described in a certain standard manner. It is important that in calculations of such displacements the point defect plays a role of a source of elastic field [40J.

According to the classification of singularities of elastic fields in an medium the defect described by the density of forces of eq. (224) is called centre of dilatation. Thus, we have used the dilatation model of an interstitial The vacancy differs from the interstitial atom by the direction of the displacement of the lattice surrounding the defect. The deformation arising is related to a _~"''''~.~ . . . . . ment of the nearest atoms towards the defect". This displacement is the symmetry of which in a simple cubic lattice may be expected to be the case of the interstitial atom. In other words, a vacancy can be .... eq. (224) but the strength of the dilatation should be considered negative It appears that the ability of the defect to play the role of the source very important in the description of the defect interaction.

* For instance, at higher temperatures the expected number of thermally activated kinks is large, and the Peierls barrier is no longer an effective obstacle.

* This "natural" direction of the displacement of the neighbours takes place in metals. In the nearest neighbours can move outwards from a vacancy.

..,."VLHJ ..' ....

iotilc.cl;~!1fl:tls

U8

A. M. Kosevich

Ch.l

The free energy of the interaction between an elastic centre of dilatation and an external elastic field is equal to [40J: (225) where eik(r 0) is the strain tensor at the position of the defect. Let the defect be displaced by an amount 0) or decreases (R < 0). The sign of R can change when the radius R varies. However, there is a particular value of R at which R = O. Thisradius, being the solution of the equation ao(R)

I

= pzz -

kTbc -3 - , a Co

A. M. Koseuich

128

Ch.l

corresponds to that shape of the helical dislocation which is in equilibrium with a given supersaturation of point defects and a fixed uniaxial load along the helix axis. Now let us assume that the tensor of external stresses has a nonzero component pyz. Since all the equations analyzed are linear, the effect of the component Pyz on the dislocation climb can be considered separately. Let a certain equilibrium radius of the helix R exist due to ()c or P~z' Then, under the action of the stress Pyz the quantity (J * acquires the addition 2

()(J

*= -

h b ) tzt y ( bi Pyz = - Pyz R cos cp,

(261)

which is written by using eq. (256). The angular dependence of eq. (261) enters the boundary conditions of eq. (257) on the dislocation tube, therefore one can predict the dependence of 1 on the angle cp. In fact, the continuity J-iv as a function of the coordinates provides its evident dependence on rp : u; o; cos tp. This means, however, that its normal derivative on the dislocation tube surface would have the same dependence on sp, Thus we have* 1

= 10 cos cp,

(262)

where the quantity 10 is proportional to Pyz and is dependent on R as well as on the characteristics of vacancies (co, D v ) ' Substitute eq. (262) in eq. (242). a 3 10 Vn = ~cos tp.

129

Crystal dislocations and the theory of elasticity

§9.5

subsequent formation of dislocation loops limiting the areas of insertion of atomic planes. The section of such a dislocation is shown in fig. 52. We shall call dislocations of this type interstitial dislocation loops. We see that the generation of prismatic dislocation loops is by growth of new atomic planes or "dissolution" of certain sections of the crystal planes. But the formation of sections of extra planes can also occur by diffusion as a result of the action of a Bardeeri-Herring source. Therefore the centre of formation of new prismatic loops can, in principle, be any dislocation line segment whose extreme points are for some reason locked and remain stationary when the shape of this section changes.

~

¢---' )----4

; ,I

r r 1 ':( r r 1 r

----

J.

J

-I I-

l---{

I--
! + bC/KC o all the vacancy loops are dissolved, and the fate of the interstitial ones depends on the ratio of their radii to the corresponding critical radius Ri', defined by the equation

dRy

It is clear that K > 0 corresponds to tension and K < 0 to compression of the crystal. We suppose for simplicity that Kbc > O. Assume that circular prismatic dislocation loops of both the types mentioned are uniformly distributed through the volume of the crystal. Let us consider an isolated dislocation loop from this system. In the analysis of its growth we shall disregard the direct interaction of the dislocations (elastic or diffusion), i.e. we shall consider each loop independently of the others. However, inasmuch as the cause of the growth of the loop is the influx of vacancies, and this influx is determined by the gradient of the concentration of the point defects near the dislocation, this necessarily gives rise to an interaction of the loops via the selfconsistent average supersaturation of point defects in the crystal. Indeed, the concentration gradient of point defects depends on the

131

Crystal dislocations and the theory of elasticity

§9.5

=

2nc oDy b log (R/r o)

[bC Co

A(R) ~J' R

(271)

Equation (271) is the basis for the consideration of the so-called coalescence of dislocation loops. Let us assume that there is a set of prismatic dislocations of the same type (vacancy ones) in the sample. When the loop dimensions are sufficiently large and the supersaturation of the vacancies is low the process of coalescence (the growth of large loops at the expense of small ones) plays the predominant role. This phenomenon is described by Kosevich, Saralidze and Slezov [84]. Acknowledgments

It is pleasure for me to express my gratitude to Professor F. R. N. Nabarro for his kind invitation to contribute to the present international collective treatise. I am deeply indebted to Lidiya Troshchenko for her invaluable help in translating the paper into English. I should also like to thank my wife Dina for her assistance in preparing the English version of the manuscript.

A. M. Kosevich

132

Ch. I

Crystal dislocations and the theory of elasticity

133

Addendum (1976) (The numbering ofsections ofthe Addendum corresponds to that ofthe corresponding sections in the main text.) 3. The direct observation of the stress fields around individual dislocations One of the simplest and most reliable methods of investigation of the stress fields around individual dislocations is that of photoelasticity. This method is very useful for the study of individual dislocation characteristics in some types of crystals which are transparent for both optical and infrared light. Internal stresses in a crystal whose optical indicatrix, when unstrained, is a sphere, initiate artificial birefringence. The birefringence can be detected by the brightening of the visual field using crossed Nicols. The intensity of plane-polarized light transmitted through the crossed Nicols depends on the difference of principal stresses acting in the plane of the wave front. I[ an edge dislocation is parallel to the direction of view and normal to the crystal surface the difference of principal stresses in the isotropic approximation equals 2Pro, where PrO is given by eq. (24). Indenbom and Tomilovskii [87] showed that the "rosette" of equal intensities in such a case is described by the following equations r

= C cos e cos 2(e

- a).

Fig. 54a

C = const.

where the polar angle e is calculated from the Burgers vector and a is the angle between the vibration direction in one of the crossed Nicols and the glide plane. When we deal with the "four-petal rosette" (at a = 45°) the glide plane is an antisymmetry plane of the birefringence field (figs. 53a, 54a). In the case of the symmetric "six-petal rosette" the glide plane is parallel to the direction of the large petal of the rosette (figs. 53b, 54b).

Fig. 53 The diagrams of intensities for double refraction fields caused by the stresses around an edge dislocation; the glide plane is horizontal: (a) the "four-petal rosette" at (j, = 45°, (b) the "six-petal rosette" at ( j , = O. Fig.54b

The photoelasticity method gives the possibility of identification of the dislocation line direction (it coincides with the microscope optical axis), glide plane, and Burgers vector sign and value. Nikitenko and Dedukh [89, 90J discussed the possibility of applying this method to the investigation of the stress field around individual dislocations in single crystals of many materials and associated local changes of physical properties of the crystals.

* This equation may

be also derived from the results obtained by Bullough [88].

Fig.54

Edge dislocations in Y3A15012 detected by the photoelasticity method: (a)

(j,

=

45°, (b) (j,

=

O.

The reader may find detailed references on photoelasticity investigations of the effect of individual dislocations on crystal properties in ref. [90J and subsequent publications.

A. M. Kosevich

134

5.4. Stress fields of pile-ups (1) Recently, a general expression for the stress field near the pile-up at any point not on the x-axis in an infinite anisotropic medium has been obtained by Solovev [91]. The stresses are expressed in the form of an analytic continuation of the distribution function into the complex plane. (2) Preininger and Bocek [92, 93J showed that the distribution function E0(x) for a double-ended pile-up in an arbitrary periodic stress field w(x) can be derived in an analytic way. 5. & 6. Dislocation group dynamics The movement of an array of discrete dislocations We consider a group of dislocations moving in the same slip plane. The dislocations are assumed to be infinitely long, straight, and parallel. If the positions of the dislocations are Xl" .. X n , then the equations of dislocation motion may be written as m*(d 2x k/dt 2 ) = bT(Xk)

+

bS(xk, t),

k

= 1, ... .n

+ L D/(xk -

xJ.

(273)

i*k

The force of inelastic origin Sex, t) can be obtained from the stress-velocity relation for an individual steadily moving dislocation: dxk/dt = V[ T(Xk)J,

135

where K is the mobility constant and the exponent m is positive but not necessarily an integer> A number of authors [95-98J have investigated such group motions by numerical methods. Some analytical similarity solutions of the non-linear equations of motion (273) and (276) were derived by Head [99]. These similarity solutions show that the arrangement of the dislocations in the group can remain of the same form, changing with time in a manner which could be described by a change in linear scale. The special case of a linear stress-velocity relation eq. (275) was discussed by Lyubov [100J, Solovev [I01J and Head [102]. Head [I02J derived some general solutions of eqs. (273) and (275) which permit an explicit description of the motion of a group of n dislocations starting from arbitrary initial positions. But these solutions are possible only under the condition that the applied stress field is, at most, a linear function of position. Zaitsev and Nadgornyi [103J considered the motion of double-ended dislocation arrays in the case when the relation (274) corresponds to the following expressions obtained experimentally:

(272)

where m* is the effective mass of the dislocation, T(X) is the total stress on the dislocation at point x, and bS(x, t) is the internal force of inelastic origin. The total stress field is the sum of the external applied stress pe(x), and the inverse stress field of all the other dislocations [see eq. (74) in the main text of this chapter] : T(X k) = pe(x k)

Crystal dislocations and the theory of elasticity

Ch.l

(274)

where VeT) is the dislocation velocity as a function of the stress on the dislocation. The stress-velocity relation, eq. (274), is often described by a linear relation similar to that of viscous friction (275) where B is the damping constant. Using eq. (272) and relation (275), Solovev [94J discussed small vibrations of dislocation arrays in inhomogeneous external fields and obtained the frequencies of those vibrations. However, in the description of the dislocation dynamics the inertial term in eq. (272) may often be neglected due to the very small value of the dislocation effective mass. In this case relation (274) where the total stress T(X) is the sum in eq. (273) acts as the equation of the dislocation motion. If a single dislocation moves with a velocity proportional to the mth power of the stress, relation (273) can be rewritten in the form (276)

parameters in this relation were taken from experiments. A continuous distribution of moving dislocations Let us write down the equations of dislocation motion in the continuum approximation. Instead of eq. (272) we can write m*(av/at) = bT(X, t)

+

(277)

bS(x, t),

where vex, t) is an average velocity of the dislocations at the point x and time t. In terms of the dislocation density E0(x, t), the total stress becomes [see eq. (75) in the main text of this chapterJ: T(X, t) = p'(x, t)

+ DP

f

E0(X" t) dx' x-x I

'

(278)

where P denotes the principal value. The system of equations (277) and (278) must be completed by the equation of continuity of the dislocation flow (156) specialized for the case of a linear dislocation distribution

aE0

---at +

a

ax (E0v) = O.

(279)

Equations (277)-(279) define the continuum approximation to the dynamics of dislocation groups, and they were earlier applied to considering the equation of motion of thin twins [104]. However, as we mentioned, the inertial term in eq. (277) may be neglected, and dislocation dynamics is usually defined by eqs. (278), (279) and the stress-velocity

136

A. M. Koseoich

Ch.l

relation. For the special case of a linear stress-velocity law these equations reduce to those considered by Rosenfield [105J. Head [106J showed that the equations (278) and (279) with an arbitrary power law stress-velocity relationship (276) possesses similarity solutions in parallel with similarity motion of the discrete group. The continuous distribution of moving dislocations for the case where the individual dislocations follow the linear stress law (275) has been widely discussed [107-111].

Dislocation description of twin dynamics Recently Boiko [110, 111J attempted to study the difficult problem of the dynamic behaviour of elastic twins by applying the dynamics of the planar dislocation pile-up. In writing the force of inelastic origin Sex) for the case of the twin one must take into consideration additional forces Ceq. (104)J in addition to the force of viscous friction Ceq. (275)]. As a result, the total force of inelastic origin has the form Sex, t) = - [So

+ ST(X, t) +

(B/b)v(x, t)].

(280)

Equations (105) and (110), where the length of the twin depends on time L = L(t), give the simplest expression for ST(X, t). The variation of the twin length with time can be defined by the quasistatic condition of eq. (107) where a(x) = p'(x, t) - (B/b)v(x, t).

(281)

Equations (278)-(281) represent a system which allows one to describe the characteristics of the twin as functions of time. Several problems on the dynamics of twins were considered [110, l l l ], namely the creation and growth of the twin under concentrated external loads and the escape of the elastic twin from an unloaded crystal. The results obtained are in agreement with the experimental data [112, 113]. 7. Localized vibrations associated with a dislocation Dislocations are peculiar defects of the crystal lattice. In the immediate neighbourhood of the dislocation atoms are considerably displaced from. their equilibrium positions, and the crystal lattice is greatly distorted. One can say that the dislocation produces local changes in the elastic properties of a crystal. The dislocation line is thus a local inhomogeneity. A simple model is a cylindrical inclusion differing from the matrix in elastic properties. Near such an inclusion, localized vibrations can appear, for which the displacement pattern is wavelike parallel to the dislocation line and decays exponentially in directions perpendicular to this line. This problem was studied by Lifshitz and Kosevich [114J and Iosilevski [115J, and the dispersion relation between the frequency of the localized mode and the wave vector component parallel to the dislocation line was found. A number of authors discussed non-linear aspects of this problem [116-118].

Crystal dislocations and the theory of elasticity

137

7.4. Radiation of elastic waves by moving dislocations Sound emission by individual dislocations Lately the problems of acoustic radiation from moving dislocations received both theoretical and experimental development. Natsik [119J studied the acoustic emission at the moment of emergence of a dislocation on to the surface of a crystal. This radiation, called transition radiation, has specific spectral and space-time properties, which allow one to distinguish it from other forms of dislocation radiation. The transition radiation of a screw dislocation consists of cylindrical shear waves which propagate into the crystal. The radiation of an edge dislocation contains longitudinal cylindrical waves in addition to transverse waves. However, the most important feature of the transition radiation from the edge dislocation is the presence of Rayleigh waves, which propagate along the crystal surface [120]. Natsik and Chishko [121J analyzed the sound emission during annihilation of a pair of straight edge and screw dislocations. Emission intensity increases rapidly with a rise in the velocity of the annihilating dislocations and is defined by the kinetic energy of the dislocations at the moment of annihilation. Dislocation radiation caused by thin walls of moving dislocations was calculated and observed by Schwenker and Granato [122]. Calculations were made on the basis of a vibrating string model which neglects dislocation interactions. The radiation was stimulated by a plane ultrasonic wave. The cylindrical waves reradiated from dislocations are all in the same phase and produce a plane acoustic wave. Since each dislocation contributes in the same way, the detected emission directly measures the motion of individual dislocations. Observation ofacoustic radiation from twinning dislocations The direct observation of sound emission from twinning dislocations was first carried out in a set of experiments by Boiko, Garber, Krivenko and Krivulya [113, 123-125]. The most favourable conditions for the observation of transition emission of sound by dislocations are obtained when an elastic twin emerges onto the surface of a crystal. As was mentioned, the elastic twin is an array of dislocations of one type, which as the external load is removed, leave the crystal under the action of the surface tension forces ST(X, t). In this case a large number of sound-emitting dislocations are accelerated to high velocities, and the acoustic pulse becomes sufficiently powerful to be detected. Figure 55 shows a typical acoustic pulse observed in such a case [123]. In the work of ref. [126J acoustic radiation during annihilation of the doubleended array of twinning dislocation pairs of opposite signs was detected.

Fig. 55 Oscillogram of an acoustic pulse generated on emergence of an elastic twin consisting of screw dislocations [123J.

138

A. M. Kosevich

Ch.l

8.4. Dislocation damping in metals In metals the spatial dispersion of the elastic moduli is caused by electrons. For this reason the dislocation dampings in normal and superconductive states of metals are different. A recent review of some problems in the theory of the dislocation damping in metals may be found in the survey by Kaganov, Kravchenko and Natsik [128]. Acknowledgment I should like to thank Dr. V. S. Boiko and Dr. V. 1. Nikitenko for providing the photographs presented in the addendum.

References [1] [2] [3] [4] [5] [6]

F. R. N. Nabarro, Theory of Crystal Dislocations (Oxford University Press, 1967). J. M. Burgers, Proc. K. ned. Akad. Wet. 42 (1939) 293. G. Leibfried, Z. Phys. 135 (1953) 23. 1. M. Lifshitz and L. N. Rozentsveig, Zh. Eksper. Teor. Fiz. 17 (1947) 783 (in Russian). J. M. Burgers, Proc. K. ned. Akad. Wet. 42 (1939) 378. M. Peach and J. S. Koehler, Phys. Rev. 80 (1950) 436. [7] E. Kroner, Kontinuumstheorie der Versetzungen and Eigenspannungen (Springer, Berlin, 1958). [8] F. R. N. Nabarro, Phil. Mag. 42 (1951) 1224. [9] W. F. Brown, Phys. Rev. 60 (1941) 139; J. S. Koehler, Phys. Rev. 60 (1941) 397. [10] A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity (Dover, 1945). [11] J. D. Eshelby, Phil. Mag. 40 (1949) 903; J. D. Eshelby, W. T. Read and W. Shockley, Acta Metall. 1 (1953) 251; A. Seeger and G. Schock, Acta Metall. 1 (1953) 519; A. J. E. Foreman, Acta Metall. 3 (1955) 322. [12] A. N. Stroh, Phil. Mag. 3 (1958) 635. [13] L. A. Pastur, E. P. Fel'dman,A. M. Kosevich and V. M. Kosevich, Fiz. Tverd. Tela 4 (1962) 2585 (in Russian); Soviet Phys. Solid St. (English Transl.) 4 (1963) 1896. [14] L. D. Landau and E. M. Lifshitz, Theory of Elasticity (Pergamon Press, Oxford, 1970). [15] A. H. Cottrell, Dislocations and Plastic Flow in Crystals (Clarendon Press, Oxford, 1953). [16] F. C. Frank, Nuovo Cim. Suppl. [10] 7 (1958) 386; W. T. Read, W. Shockley, Imperfections in Nearly Perfect Crystals (Wiley, New York, 1952) p. 352. [17] J. Weertman, Phil. Mag. 11 (1965) 1217. [18] F. Kroupa, Czech. J. Phys. B11 (1961) 847. [19] F. C. Frank, Rept. Conf. Defects in Crystalline Solids (The Physical Society, London, 1955) p. 159. [20] Dislocations and Mechanical Properties of Crystals (Wiley, New York, 1957). [21] V. L. Indenbom and G. N. Dubnova, Fiz. Tverd. Tela 9 (1967) 1171 (in Russian); Soviet Phys. Solid St. (English Transl.) 9 (1967) 915. [22] J. C. M. Li, Acta Metall. 8 (1960) 296; 9 (1961) 384. [23] Imperfections in Nearly Perfect Crystals (Wiley, New York, 1952). [24] Dislocations in Metals (AJ.M"E., New York, 1954). [25] J. D. Eshelby, F. C. Frank and F. R. N. Nabarro, Phil. Mag. 42 (1951) 351. [26] A. N. Stroh, Proc. Roy. Soc. A 223 (1954) 404. [27] G. Leibfried, Z. Phys. 130 (1951) 214. [28] N.1. Muskhelishvili, Singular Integral Equations (Noordhoff, Groningen, 1953). [29] R. E. Peierls, Proc. Phys. Soc. 52 (1940) 34. [30] F. R. N. Nabarro, Proc. Phys. Soc. 59 (1957) 256. [31] F. C. Frank and W. T. Read, Phys. Rev. 79 (1950) 722.

Crystal dislocations and the theory ofelasticity

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140 [72] [73] [74] [75] [76] [77]

A. M. Kosevich

Ch.1

G. Leibfried, Dislocations and Mechanical Properties of Crystals (Wiley, New York, 1957) p. 495. T. A. Kontorova and Ya. I. Frenkel, Zh. Eksper. Teor. Fiz. 8 (1938) 89, 1340, 1349 (in Russian). J. D. Eshelby, Proc. Roy. Soc. A266 (1962) 222. J. S. Koehler, Imperfection in Nearly Perfect Crystals (Wiley, New York, 1952) p. 197. A. Granato and K. Lucke, J. Appl. Phys. 27 (1956) 583. A. M. Kosevich and V. D. Natsik, Zh. Eksper. Teor. Fiz. 51 (1966) 1207 (in Russian); Soviet Phys. JETP (English Transl.) 24 (1967) 810. [78] A. A. Slutskin and G. G. Sergeeva, Zh. Eksper. Teor. Fiz. 50 (1966) 1649 (in Russian); Soviet Phys. JETP (English Transl.) 23 (1966) 1097. [79] A. M. Kosevich and V. D. Natsik, Fiz. Tverd. Tela 10 (1968) 1545 (in Russian); Soviet Phys. Solid St. (English Transl.) 10 (1968) 1220. [80] J. Friedel, Dislocations (Pergamon Press, Oxford, 1964). [81] B. A. Vershok and A. L. Roitburd, Fiz. Metal. Metalloved. (USSR) 32 (1971) 269 (in Russian); Phys. Metals Metallog. (English Transl.) 32 (1971) 43. [82] F. R. N. Nabarro, Phil. Mag. 16 (1967) 231. [83] A. L. Roitburd, Fiz. Tverd. Tela 7 (1965) 1142, 1349 (in Russian); Soviet Phys. Solid St. (English Transl.) 7 (1965) 916, 1089. [84] A. M. Kosevich, Z. K. Saralidze and V. V. Slezov, Fiz. Tverd. Tela 6 (1964) 3383 (in Russian); Soviet Phys. Solid St. (English Transl.) 6 (1965) 2707. [85] A. M. Kosevich, I. G. Margvelashvili and Z. K. Saralidze, Fiz. Tverd. Tela 7 (1965) 464 (in Russian); Soviet Phys. Solid St. (English Transl.) 7 (1965) 370 [86] A. M. Kosevich, Z. K. Saralidze and V. V. Slezov, Zh. Eksper. Teor. Fiz. 50 (1966) 958 (in Russian); Soviet Phys. JETP (English Transl.) 23 (1966) 636. [87] V. L. Indenbom and G. E. Tomilovskii, Kristallografiya 2 (1957) 190 (in Russian); Soviet Phys. Crystallogr. (English Transl.) 2 (1957) 183; Dokl. Akad. Nauk SSSR 115 (1957) 723 (in Russian). [88] R. Bullough, Phys. Rev. 110 (1958) 620. [89] V. L Nikitenko, L. M. Dedukh, S. Sh. Gendelev and N. G. Shcherbak, ZhETF Pis. Red. 8 (1968) 470 (in Russian); JETP Letters (English Transl.) 8 (1968) 288. [90] V. 1. Nikitenko and L. M. Dedukh, Phys. stat. sol. (a) 3 (1970) 383. [91] V. A. Solovev, Phys. stat. sol. (b) 65 (1974) 857. [92] D. Preininger and M. Bocek, Mater. Sci. Eng. 12 (1973) 131. [93] D. Preininger, Mater. Sci. Eng. 21 (1975) 77. [94] V. A. Solovev, Fiz. Metal. Metalloved. 34 [4](1972) 836 (in Russian); Phys. Met. Metallogr. (English Transl.) 34 [4] (1972) 153. [95] J. Weertman, J. Appl. Phys. 28 (1957) 1185. [96] A. R. Rosenfield and G. T. Hahn, Dislocation Dynamics (McGraw-Hill, New York, 1968, p. 255); Acta Met. 16 (1968) 755. [97] M. F. Kanninen and A. R. Rosenfield, Phil. Mag. 20 (1969) 569. [98] A. R. Rosenfield and M. F. Kanninen, Phil. Mag. 22 (1970) 143. [99] A. K. Head, Phil. Mag. 26 (1972) 43. [100] B. Ya. Lyubov, Dokl. Akad. Nauk SSSR 152 (1963) 1092 (in Russian); Soviet PhysDokl. (English Transl.) 8 (1964) 1007. [101] V. A. Solovev, Fiz. Metal. Metalloved. 33 [4](1972) 690 (in Russian); Phys. Met. Metallogr. (English Transl.) 33 [4] (1972) 16. [102] A. K. Head, Phil. Mag. 26 (1972) 55. [103] S. I. Zaitsev and E. M. Nadgornyi, Phys. stat. sol. (a), 9 (1972) 353. [104] A. M. Kosevich, Teoriya Dislokatsii, ch. 4. Dislokatsionnaya teoriya tonkikh dvoinikov (Theory of Dislocations, Part 4, Dislocation theory of thin twins) (Preprint FTI AN USSR, Kharkov, 1963). [105] A. R. Rosenfield, Phil. Mag. 24 (1971) 63. [106] A. K. Head, Phil. Mag. 26 (1972) 65 [107] A. K. Head and W. W. Wood, Phil. Mag. 27(1973) 505, 519. [108] A. K. Head, Phil. Mag. 27 (1973) 531 [109] W. W. Wood and A. K. Head, Proc. Roy. Soc. A 336 (1974) 191. [110] V. S. Boiko, Phys. stat sol. (b) 55 (1973) 477.

Crystal dislocations and the theory of elasticity

141

[111] V. S. Boiko, Dinamika Dislokatsii (Dislocation Dynamics) (Naukova Dumka, Kiev, 1975, p. 161). [112] V. S. Boiko, R. I. Garber and V. F. Kivshik, Fiz. Tverd. Tela 16 (1974) 591; 17 (1975) 3655 (in Russian); Soviet Phys. Solid St. (English Transl.) 16 (1974) 384; 17 (1975) 2376. [113] V. S. Boiko, R. I. Garber and L. F. Krivenko, Fiz. Tverd. Tela 16 (1974) 1451 (in Russian); Soviet Phys. Solid St. (English Transl.) 16 (1974) 930. [114] I. M. Lifshitz and A. M. Kosevich, Rep. progr. phys. 29 (1966) 217. [115] Ya. A. Iosilevskii, Fiz. Metal. Metalloved. 30 (1970) 701 (in Russian); Phys. Met. Metallogr, (English Transl.) 30 [4] (1970) 29. [116] A. A. Maradudin, Fundamental Aspects of Dislocation Theory (NBS Special Publication 317 V.I., 1970, p. 205. [117] V. K. Tewary, J. Phys. C7 (1974) 261. [118] I. M. Dubrovskii and A. S. Kovalev, Fiz. Nizkikh Temp. 2 (1976) 1483 (in Russian), Soviet J. Low Temp. Phys. (English Transl.) (1977) 2, 726. [119] V. D. Natsik, ZhETF Pis. Red. 8 (1968) 324 (in Russian); JETP Letters (English Transl.), 8 (1968) 198. [120] V. D. Natsik and A. N. Burkanov, Fiz. Tverd. Tela 14 (1972) 1289 (in Russian); Soviet Phys. Solid St. (English TransL) 14 (1972) 1111. [121] V. D. Natsik and K. A. Chishko, Fiz. Tverd. Tela 14 (1972) 3126 (in Russian); Soviet Phys. Solid St. (English TransL), 14 (1973) 2678. [122] R. O. Schwenker and A. V. Granato, Phys. Rev. Lett. 23 (1969) 918; J. Phys. Chern. Solids 31 (1970) 869. [123] V. S. Boiko, R. I. Garber, L. F. Krivenko and S. S. Krivulya, Fiz. Tverd Tela 12 (1970) 1753; 15 (1973) 321 (in Russian); Soviet Phys. Solid St. (English Transl.) 12 (1970) 387; 15 (1973) 238. [124] V. S. Boiko, R. I. Garber, V. F. Kivshik and L. F. Krivenko, Zh. Eksper. Teor. Fiz. 71 (1976) 708 (in Russian). [125] V. S. Boiko, R. I. Garber and L. F. Krivenko, Fiz. Tverd. Tela 16 (1974) 1233 (in Russian); Soviet Phys. Solid St. (English TransL) 16 (1974) 798. [126] V. S. Boiko, R. I. Garber and L. F. Krivenko, Dinamika Dislokatsii (Dislocation Dynamics) (Naukova Dumka, Kiev, 1975, p. 172). [127] V. A. A1'shitz and V. L. Indenbom, Usp. Fiz, Nauk 115 (1975) 3 (in Russian); Soviet Phys. Usp, (English Transl.) 18 (1975) 1. [128] M. I. Kaganov, C. Ya. Kravchenko and V. D. Natsik, Usp. Fiz. Nauk 111 (1973) 655 (in Russian).

CHAPTER 2

Dislocations in Anisotropic Media J. W. STEEDS and J. R. WILLIS H. H. Wills Physics Laboratory University ofBristol, UK and School of Mathematics University of Bath, UK

© North-Holland Publishing Company, 1979

Dislocations in Solids Edited by F. R. N. Nabarro

Contents 1. Introduction 145 2. Fundamental equations 146 3. Three-dimensional problems 147 3.1. Fourier transform formulation 147 3.2. The straight line segment 149 4. Two-dimensional problems 154 4.1. The stress function method 154

1. Introduction

4.2. General simplifications 155 4.3. Effects of crystal symmetry 156 5. Applications of anisotropic theory 159 5.1. Dislocation pinning 160 5.2. Glide and climb of edge dislocations 161 5.3. Glide and cross slip of screw dislocations 162 5.4. Dislocation interactions 163 5.5. Slip systems 163 6. Microscopic measurement of elastic properties 163 6.1. Dislocation bends 163 6.2. Equilibrium angle of dislocation dipoles 6.3. Image widths in the electron microscope References

164

163 164

Almost all real crystals are elastically anisotropic and properties of materials that are influenced by dislocation stress fields should really be studied within the framework of anisotropic elasticity theory. Isotropic theory involves fewer parameters and so is both better known and easier to apply, and has been of tremendous value in the development of physical theories which involve knowledge of dislocation stress fields. Anisotropic theory has now reached a state of development at which its application is feasible in a wide range of situations. In particular, new three-dimensional solutions have been evolved whose application is made routine by present-day computing facilities, although this would not have been the case ten years ago. Also, various workers have been motivated to study practically important situations in which the anisotropic theory simplifies to a point where purely analytic calculations are possible. This chapter contains a brief review of both of these developments, slanted inevitably towards the work of the authors, but still, hopefully, presenting an honest picture of the field at the present time. Three-dimensional theory has been advanced particularly by Lothe [lJ, Brown [2J, and Indenbom and Orlov [3, 4J, who used very ingenious similarity arguments to express three-dimensional fields in terms of two-dimensional ones which, however, were not immediately available in sufficiently explicit form. Willis [5J proposed an alternative approach, based upon the use of the Fourier transform, which yielded solutions that were completely explicit, and a variation of this approach, which yields both the representations of Willis [5J and a new representation which may be more useful for computations, is outlined in sect. 3 below. Simple analytic results whose application calls for little or no computation are available for a great many two-dimensional problems which involve suitably oriented straight dislocations. Although these are contained in principle in the general threedimensional formulation they are, of course, best obtained directly, and we outline briefly an approach whose consequences are developed in detail in the book by Steeds [6J. This chapter would be redundant, or at least disappointingly restricted in its use, if it were to emerge that anisotropic theory did no more than provide small corrections to results obtained using isotropic theory. This'is certainly not the case, however, and we sketch briefly some applications in which it is easy to recognise that predictions based on anisotropic theory may be qualitatively different from those implied by isotropic theory. The consequences of the use of anisotropic theory have by no means been exhausted, and further significant applications may be expected to appear in the future.

J. W. Steeds and J. R. Willis

146

Ch.2

2. Fundamental equations We introduce here the notation that will be followed throughout this chapter. The elastic displacement of a point whose Cartesian components are Xi (i = 1, 2, 3) will be denoted by a vector whose Cartesian components are ui(x) (i = 1,2, 3), and we define the components ciix) of the infinitesimal strain tensor as

(I)

The stress in the body has components (Jij(x), which are related to the strain components of eq. (1) through the generalized Hooke's law (2) with inverse

=

(3)

Sijkl(Jkl·

In eqs. (2) and (3) the summation convention is adopted, that the use of a repeated suffix implies summation over suffix values 1, 2, 3. The elastic moduli Cij k1 are the components of a fourth order tensor, as are the elastic compliances Sijkl. As the stress and strain are each symmetric tensors, we may take

147

convenient abbreviation but it should be noted that c mn' smn are not the components of any second order tensor and that their transformation law under a change of coordinates must be obtained by referring back to the tensors Cijkl' Sijkl from which they were derived. The equations which govern the displacement field are the equilibrium equations o(JiiOXj

,,/x) = i (:~; + :;,}

Cij

Dislocations in anisotropic media

§2

=

o.

(9)

Two approaches are possible, which have advantages in different situations. The first, which is exploited in sect. 3, is to substitute eqs. (1) and (2) into eq. (9) to obtain the equations Cijkl(02Uk/OXj OX l)

= 0

(10)

for the displacements themselves. The second, which is useful particularly for twodimensional problems, is to relate the stresses to stress functions, which are then restricted so that the stresses are indeed derivable from a displacement field uJx); this is taken up in sect. 4.

(4)

and similarly for

Sijkl.

It is usual to assume also that

3. Three-dimensional problems (5)

in which case the relation of eq. (2) is derivable from a potential energy function in the form

oW

(Jij

= ;:;--;

(6)

uCij

but many of the results of this chapter, including all those of sect. 3, are independent of this property. It will be convenient later to use an alternative notation, which exploits particularly the symmetries ofeq. (4). In this notation, the replacements 11 ~ 1, 22 ~ 2, 33 ~ 3, 32 ~ 4, 31 ~ 5, 21 ~ 6 are made, and we define eij

= =

Cij 2cij

for i = j, for i ::f.: j,

Cij kl Sijkl 2S ij kl 4S ijkl

= = = =

Cmn Smn Smn Smn

for for for for

all m, n, m, n ~ 3, m or n > 3, m and n > 3.

(7)

The eqs. (2) and (3) expressing Hooke's law then become (8) where here, exceptionally, to avoid long expressions, the summation convention has been employed for indices taking the values 1 to 6. The two-suffix notation is a

3.1. Fourier transform formulation This section is devoted to the representation of the stress field of an arbitrary dislocation loop in an infinite elastic body. The dislocation is defined by imposing upon the body a discontinuity of displacement across a specified surface S. The elastic displacement field ui(x) so generated satisfies the equations of equilibrium (10), together with the conditions [uJ

=

hi'

[(JijnjJ

= 0

(11)

across S, where [uJ denotes the difference between U i on the positive side of S and on the negative side of S, the Burgers vector has components hi and n, is the normal to S, taken from the negative side to the positive side. Now it was shown by Volterra [7J that the displacement ui(x) could be represented in terms of the stresses produced by unit point body forces, so that u,(x) =

Is bjP)k(X' -

x)nk(x') dS(x'),

(12)

where p~k(XI - x) is the Uk) component of stress at x', produced by a unit point body force applied in the i-direction at x. It is well known that, although the displacement suffers a jump across S, the elastic distortions ou)ox o> and hence the stresses, are continuous everywhere except at the boundary oS of S, so that, from a mathematical point of view, the choice of S itself is unimportant. This has been demonstrated by

Ch.2

§3.1

au)axp that results from differentiating

and

J. W. Steeds and J. R. Willis

148

Mura [8], who transformed the expression for eq. (12) using Stokes' theorem, to obtain

auJax p = -

J

'ts

8 pj[bmp im(x

l -

Dislocations in anisotropic media

149

(24)

x) dx).

(13)

The corresponding result for a continuous distribution of dislocations, which contains eq. (13) as a special case, was derived directly by Willis [9]. Equation (13) forms the starting point for the work of the present section. To obtain a representation for P~m(x), we note that (14)

where the "Green's functions" Gir(x) satisfy the equations

I'll

= 1. where dco is an element of the unit sphere, The integration with respect to I~I can now be performed, in the sense of distributions, to give

aU = -8 i s-,» b me[mrs rrm -a i

Xp

tt

p,

e--> 0

dx j I

es

f

i

1,,1

= 1

gsir('I) [ .( '_dw) _ . ]2' 'I x X 18

(25)

The integral with respect to x', depends upon the choice of as. Willis [5] gave formulae, particularly for straight segments and for elliptical loops. The more generally useful result is that for a straight segment, since the distortions due to any polygonal loop can be generated from it, and the further reduction of the integral of eq. (25) is illustrated for this important case below.

(15) 3.2. The straight line segment

Equations (15) are now readily solved by taking Fourier transforms, to give (16)

x = ae

where D(~)

Although as must necessarily be closed, it is nevertheless possible to obtain an expression for the contribution to auJaxp from the straight line segment

= IL(~)I,

Nir(~)

= [Adj

and the matrix

(17) L(~)lr

L(~)

= [L -l(~)lrD(~)

(18)

has components

It may be noted that Nir(~) is homogeneous of degree 4 in ~, D(~) is homogeneous of degree 6, and D(~) = 0 only when ~ = 0, if all three components of ~ are real, since the equations (10) must be elliptic. Equation (13) now gives

8~3 8PilbmClm"t,s d.x;

t(fJ - ae),

o~

fff

d';, d';2

d';3g'i.(~) exp [-i~·(x' -

x)], (20)

where (21) and the task is now to reduce the integrals. One integration can be performed immediately, exploiting the homogeneity of gsir(~)' by transforming to spherical polar coordinates so that gsir(~) = gsir('I )/I~I,

(22)

'I = ~/I~I

(23)

t

~

1,

(26)

from which an expression for a general polygonal loop can be developed. Replacing as by the segment of eq. (26), then, gives dx, = (f3j - Ct.) dt,

'I' (x' - x) = 'I·(ae - x) (19)

-

+

+ t'l'(fJ -

ae)

(27)

and integration with respect to t now gives

au. = -a I

xp

i 8n

-3 8

pj[bmC[mrs(!3j -

(I)

lim e->O

1 [1

fi

1,,1

dwgsir('I) = 1

1 ]

x 'I' (fJ - ae) 'I' (ae - x) - is - 'I' (fJ - x) - is .

(28)

The singularity that occurs when 'I' (fJ - ae) = 0 is only apparent as the term in square brackets vanishes, but there remain the points, 'I = ±n say, at which all of 'I' (fJ - x) and 'I' (fJ - ae) vanish and the integrand becomes of order 1/82 . To cope with this, we write (29)

for which the term in curly brackets vanishes at 'I = ± n. The integral of the last term is completely explicit and some tedious analysis which will not be reproduced, shows that it tends to zero as 8 tends to zero. Hence, gsir('I) in eq. (28) can be replaced by the term in curly brackets above. Now since this function is odd, replacing 'I by - 'I in the integral shows that au)axp is its own complex conjugate and hence is real, as it should

Ch.2

J. W. Steeds and J. R. Willis

150

be. A quick way of evaluating one more integral is therefore to identify the right side of eq. (28) (with the modified form of gsir(II)) with its real part, and to note that

§3.2

Dislocations in anisotropic media

we have d(

ds = P7t {

i .} q .(oe - x) - 18

----+

-nb[q.(oe - x)J

as

8 ----+

0,

(30)

QXp

= _1_

({3. _ 8n 2 8PJI mClmrs J .b

.) { ltJ

1

!

IfJ - xl JL p

_ _I_I- gsir(q) - (q·n)gsir(n) loe - xl JL a n- (fJ - oe)

g",(q) - (q. n)g",(n) ds

II' (fJ

- oe)

dS}'

(31)

II'(oe - x) = 0

(32)

and L p is defined similarly. The expression of eq. (31) can be verified by strict analysis. It is useful in that the integrands are finite, containing only apparent singularities, and may therefore have some advantage over the corresponding formula (4.19) of Willis [5J, which was explicit but involved finding the roots of a sextic equation. Contour integrals of this general type have been developed for infinite straight dislocations by Barnett and Swanger [lOJ, who claim this advantage on the basis of worked examples. The original formula (4.19) of Willis [5J may be recovered from eq. (31) by evaluating the contour integrals. A way of achieving this is to transform the integrals over La and L p by projection. For example, the integral over La may be projected onto the pair of lines Ca: I~

X

nl =

1,

~·(oe

- x) = 0,

=

mz'

(35)

while

(36)

and it may be noted that ~ '(fJ - oe) is independent of ( on Ca' Thus, the integral over La is transformed into fa

where La is the contour

1111 = 1,

d(

1 + (2

gsirCq) - (q·n)gsir(n) _1~12{gsir(~) - (gsir(n)/1~12} 11' (fJ - oe) ~. (/1 - oe)

and similarly for [11' (fJ - x) - ieJ -1. It then follows that QUi

151

=

Jera ~'(/1 1-

{(J:) (gsir(n)} oe) gsir ~ - 1 + (2 d(.

Further, as the integrand is an even function of~, the contribution is the same from each branch of Ca' Hence, f

2

a=

foo

d(

_ 00 (/1 - oe)· q(oe)

{(J:)

(gsir(q)}

9 sir ~ - 1

+ (2

(37)

'

where l1(oe) is the point of tangency

q(oe) =

n x (oe - x)/Ioe - xl.

(38)

The integrand in eq. (37) is 0(1/(2) as ( tends to infinity and so the integral can be evaluated by closing the contour in the upper half of the complex (-plane and using Cauchy's theorem. This gives fa

4ni { 1} = (fJ - oe). q(oe) hir[q(oe)J - 2gsir(n) ,

(39)

(33)

where

which are tangent to La and parallel to n, as shown in fig. 1. If we set ( = ~'n,

(34) CC(

~ N N QD hir(l1) = N~f~sNir(~ )/ [ nkQ~k ~N

«NJ) ,

(40)

= 11 + n(N

(41)

and (N (N = I, 2, 3) are the roots ( of the equation

+ nO =

D(l1

0

(42)

that lie in the upper half-plane. Hence, upon noting that

loe - xl(/1 - oe)·q(oe) = 1/1 - oelp,

(43)

where p is the perpendicular distance of x from the dislocation segment, substitution of eq. (39), and the corresponding expression from L p , into eq. (31) yields QUi _ :::l

uX p

Fig. 1 The curves La and ca' The vector (Ot - x) is normal to the plane of the page.

-

i {3j - ltj f 2tt epjlbmClmrs p 1/1 - oe I lhir[q(oe)J

-

as given by Willis [5J.

_

}

hir[q(/1)J,

(44)

Ch.2

J. W. Steeds and J. R. Willis

152

It may be noted that, if at and fJ tend to infinity in opposite directions, then q(at) tends to m, and q(fJ) tends to - m, where m is perpendicular to both n and the dislocation segment. In this case, eq. (44) reduces to

. (45) upon use ofthe easily verified relation thathir(m) and!sir( - m) are complex conjugates. Equation (45) gives an explicit representation for the distortions due to an infinite straight dislocation. Indenbom and Orlov [3J have given a representation for ou)ox p which involves both the distortions due to infinite straight dislocations and their derivatives with respect to orientation. This representation is rather more cumbersome to apply than the one above and the relationship between the two representations has not yet been clearly worked out. However, Indenbom and Orlov [3] also give a simpler expression, embodied in their fig. 4, for the distortion due to a line of body force. Equation (13) represents the dislocation field in just this way, so that this representation is applicable to dislocations, though this was not noted by Indenbom and Orlov. It is less explicit but is exactly equivalent to eq. (44). The representation that Indenbom and Orlov proposed for dislocations was based essentially upon building polygonal loops from angular dislocations, as these do not display the problem that they cannot exist by themselves. This work is further developed and applied in references [11-13]. The same approach was adopted by Lothe [1]. It is of some interest to evaluate the distortions ou)oXp for points close to the dislocation segment. To this end, it is convenient to suppress irrelevant suffixes and to write eq. (44) in the form

~Ui uX

=

p

~ lj{h/q(at)) p

E(8)

=

ljh/m)

lp

(51)

as follows. Choose axes so that the arm (1) of the angular dislocation lies along the x 1-axis, while the arm (2) lies in the x CX2 plane, at an angle 8 to the x 1-axis, as shown in fig. 2. For the arm (1), I = (1,0,0), while for the arm (2), I = (cos 8, sin 8, 0). The

p (2)

o

(1)

e

The angular dislocation. The force

x, (J

is evaluated at p, distant r from O.

(47)

force a experienced by the arm (2), at the point distant r from the bend, is composed of the self-force obtained from eq. (50), with at = 0, m = (sin 8, -cos 8, 0) and 1111-+ CfJ, and the force produced by the dislocation (1), which is obtainable from eq. (46) with q(at) = (0, -1,0), q(P) = m andp = r sin 8. Thus,

and I is a unit vector parallel to (fJ - at). The definition of eq. (47) is permissible since oujox p is necessarily real. It can be verified that hj(q) is an odd function of n. Now when x is close to the dislocation segment, but not near either end,

+ Ix-at I'

153

The first term on the right side of eq. (50) is the field due to an infinite dislocation' the second term contains the derivative with respect to line orientation of the function him). An infinite straight dislocation by definition exerts no force upon itself and so the force experienced by a finite segment contains a contribution from itself which comes from the second term on the right side of eq. (50). This is consistent with the findings of Lothe [1] (and also Brown [2, 14J and Indenbom and Orlov [3J), on the force experienced by either arm of an angular dislocation. In fact, the stress component, a say, responsible for the force on the dislocation is a linear combination of the distortions and so also has the form of eq. (50). It may be expressed in terms of the "energy factor"

Fig. 2

= -!ncpj/bmc/mrs5 {hir(q)}

q(et,) '" m

Dislocations in anisotropic media

(46)

- h/q(fJ))} ,

in which h/q)

§3.2

q(fJ) '" -m

o

dh2(m)} 1 {E(O) - h (m)} + -1 { cos 8 _1_ dh (m) + sin 8 _ = -'-8 _ . 1

r sin

r

d8

d8

(52)

But

lp

+ --.

(48)

Ix - fJl

dE dh 1 d8 = cos 8 dB

+

dh z . sm 8 dB

-

. sm 8 h ,

+

cos 8 hz,

(53)

Equation (46) now gives, asymptotically,

~Ui

oX

1 '"

{

p ljhj(m)

p

+

p

oh/m) p Ix _ atl 1k om - h/ -m) - Ix _ k

PI

oh/m)} 1k om . k

and substitution of eq. (53) into eq. (52) gives (49)

Hence, as him) is an odd function of m, bu, oX

p

2

'"

pljh/m) +

{II}

Ix - atl -

Ix

PI

oh/m) l)k om . k

(50)

a = 1 {~(08) _ cot 8 E(8) r sin

+

dE(8)},

d8

(54)

in agreement with Lothe [IJ, Brown [2J and Indenbom and Orlov [3]. The equilibrium configuration of an angular dislocation can be found by requiring that the force on either arm is zero; Lothe [IJ and Indenbom and Orlov [3] showed

J. W. Steeds and J. R. Willis

154

Ch.2

that the predicted equilibrium positions are consistent with minimizing the energy of a dislocation loop of given area, using the "line tension" approximation and Wulff's theorem, even though there is no rigorous a priori justification for this procedure. Indenbom and Dubnova [12J have shown, however, that the "line tension" approach gives incorrect predictions for triple nodes, etc. The relation between the "line tension" approximation and the exact energy of dislocation loops has been further examined by Brown [14]. Such energy calculations make no allowance for the dislocation "core", a finite result being obtained by arbitrarily defining a cut-off radius about the dislocation line, and ignoring the contribution from within it. In applications to small dislocation loops, however, the "line tension" term does not dominate and then some more rigorous allowance for the dislocation core is essential. An allowance which can be made is to allow for the work done by the tractions at the core boundary, ascalculated from the continuum solution; this at least is feasible and ensures that the energy calculated is independent of the choice of "cut" across which the displacement is allowed to jump. Bacon et al. [15J calculated the energy of rhombus-shaped dislocation loops in cubic crystals on this basis, and successfully explained the observed configurations in quenched aluminium. The "core" correction was essential in this application. Of course, for the smallest loops, a significant part of the energy is contained in the core itself and to assess this component detailed models are required, discussion of which falls within the scope of ch. 5.

§4.l

Dislocations in anisotropic media

155

and

where F; == aF/ax l '

F; == aF/ax 2 etc.

and Sij are the reduced compliances defined by

Solutions of eqs. (57) and (58) will have the form [16J

F = fJlBngn(zn),

(59)

= fJlCngn(zn),

(60)

a.

The use of n - 8 1 instead of 8 1 ensures that the displacement has the common value zero on both sides of the interface where it intersects the x-axis. With eq. (6) continuity of the traction II ow/or at the interface gives

1l2(f3 -

(14)

This is only an approximation, but it is quite easy to find the exact displacement and energy when the singular line is excluded by a stress-free circular hole, not necessarily small. In fact, since eq. (6) is satisfied on Co as well as C 2 the solution (11) also leaves Co stress-free, and its interior can be removed without upsetting the displacement. Imagine that the material is slit along the x-axis between Co and C 2 and that the two sides of the cut are given a relative displacement b so as to generate the dislocation. The work which must be done in this process, and thus the energy of the dislocation, is given by

y

a2

')I)

= 1l1(f3' -

')I').

With 8 eliminated by eq. (5) continuity of w at the interface gives

ub 2 a 2 - ~~ -In . 4n aao

f

-

~i

= a'8 + f3'8 2 + ')I'(n - fJ 1 ) ,

2

lIb E = In (a 2 4n

~i

Here we cannot give a definite value to the additive constant, since the total energy is formally infinite. Suppose next that the shear modulus is 112 inside the circle r = a and III outside it. The method of images can still be applied [6]. We take w to be a general linear combination of 8 1 , 82 and 8 in both regions, say

This may be regarded as the negative gradient

Fx =

175

for quite generally it makes no sense to ask whereabouts within its hollow core a dislocation actually lies ([5J p. 570). In fact the difference of the elastic fields for two such supposedly different positions would be characterised by zero traction on the hole and a single-valued displacement round it, and would thus, in the absence of applied loads, be zero. For a o « a eq. (15) agrees with eq. (14). We find similarly that for a dislocation in the matrix near a hole the image force due to the terms in fJ and fJ 2 in eq. (12) is given by F; = -oE/a~l with

1)

lIb

Boundary problems

§2.1

-

+

~o - ao), a o - ~2)

(15)

which can be expressed entirely in terms of ~o or ~2 with the help of eqs. (3) and (9). Either ~o or ~2 is acceptable as a parameter specifying the position of the dislocation,

f3

+

')I

= f3' +

a

')I',

+

+

f3 = a'

f3'.

If the dislocation is in the cylinder at P 2 we must have a = 0, f3 = 1, ')I' = 0 which gives

+ K(n - 8 1 ) , = (K + 1)82 - K8,

2nw/b = fJ 2

r


a,

(18)

with K

= (1l2 - 1l1)/(1l2 +

(19)

/11)'

If the dislocation is in the matrix at P 2 we must put a = 0, f3 = 0, gives [7J 2nw/b = (K - 1)(t1 1

-

= K(fJ2

+

-

8)

n),

81

-

r


a.

(20)

Apart from an additive constant the image displacement, and consequently also the image term near P 2 in eq. (18), is K times what it was in eq. (11), so that the energy corresponding with eq. (14) is /1

b2

E = ~n KIn (a 2

-

~D

+

canst.

(21)

J. D. Eshelby

176

Ch.3

Similarly, by comparing eq. (20) and eq. (12) we find from eq. (16) that for the dislocation at P 1 in the matrix the energy is /1

E = -~ Kin 4n

e- a 1 ;C2

2

+

const.

(22a)

':>1

With the notation of fig. 3 the displacements in eqs. (11), (12), (18) and (20) apply equally to a dislocation near a plane interface [8]. To adapt eq. (14) to this case replace a 2 by ~ 1 ~ 2' ~ 1 ~ 2 by 2~ and note, from fig. 2a that a, ~ l' ~ 2 become equal for large a. This gives

E = /1b In 2~ 4n ao 2

2.2. Screw dislocations in cylinders of general cross-section In this section we shall extend the analysis of the last section to screw dislocations in cylinders of other than circular cross-section with stress-free surfaces. As a justification for doing so we recall that crystal whiskers of, for example, hexagonal or rectangular cross-section have been discovered or grown intentionally, and that in some of them there is an axial dislocation which produces interesting effects. What happens when a finite rod is cut from the infinite cylinder envisaged in the theory of anti-plane strain is discussed in sect. 3.4. The harmonic anti-plane displacement w may be considered to be the real part of an analytic function of the complex variable z = x '+ iy, say

(22b)

w

for a dislocation distant ~ from the stress-free surface of a semi-infinite solid (fig. 3). y

177

Boundary problems

§2.2

+ icp =

g(z).

The potential cp conjugate to w is in many respects more useful than w itself [8]. They are related by the Cauchy-Riemann equations

ow oy

ocp _ pzy

- ox

r

(25)

;

p

which state that the gradients of wand ip are orthogonal. Hence the condition 0 on a free surface becomes cp = const. on a free surface. The expression for the force exerted on a screw dislocation by an applied elastic field,

ow/on =

becomes a simple gradient in terms of tp : F = - ub grad cp.

Suppose that in addition to the x-y plane we have a second plane with Cartesian coordinates ~, 11. Then

Fig. 3 A dislocation near a plane interface.

Cpl:, -

The accurate expression, eq. (15), gives

/1b 2 ;c /1b 2 ;c E = sinh - 1 ~ = cosh - 1 ~ 4n ao 4n ao

2

/1 b ~n K In 2~

+ const.

iw(, =

:n

In ,

or (23)

where 2~ is the object-image distance, whereas ~c is the distance of the centre of the hole of (possibly large) radius ao from the free surface. The relation between eqs. (14) and (21) persists in the limit, and so E =

(26)

(24)

is the energy associated with a screw dislocation at a distance ~ from the surface of a semi-infinite solid of shear modulus /12 which is bonded to a second semi-infinite solid of shear modulus /11'

cp

(,

b 2n

b 2n

= --In 1'1 = --In (~2 + 11 2 ) 1 / 2 ,

w = -b tan -1 11-, (, 2n ~

(27)

with ( = ~ + i11 represent a screw dislocation at ( = 0 with the unit circle stress-free, since cp(,

= 0 or

1(1 =

1

(28)

1.

The suffix (is a reminder that a quantity is considered as a function

1(1 =

of~,

11, not x,y.

178

Ch.3

J. D. Eshelby

Let us relate x, y to

~,

11 by

( = fez),

(29)

where fez) is analytic. We can then regard cp as a function of x, y, in which case we drop the suffix:

cp(x, y) =

cp,[~(x,

02cp, _ 0

O~2 + Ot/2 -

179

If the singularity-free transformation ( = fez) maps the interior of C onto the interior of the unit circle in the (-plane so that

If(z)12 = f(z)f(z) = 1 on C

(32)

cp _ iw = ..': In I - f(z')f(z) 2n fez) - fez')

(33)

then

represents a screw dislocation at x', y'(z' = x' + iy') in the cylinder with stress-free contour C. (The bar denotes the complex conjugate.) To verify this, note that by eq. (32) fez) may be replaced by l/f(z) on C, which changes the argument of the logarithm in eq. (33) into the reciprocal of its own complex conjugate, so that its modulus is unity, and that the harmonic function

satisfied by cp" being invariant under a rotation or expansion of the coordinate system, becomes

02cp ox 2

Boundary problems

Y), nt», y)].

If we write x + iy = f- 1(0 we have the Cauchy-Riemann relations OX/O~ = oy/ot/, ox/ot/ = - oy/o~ analogous to eq. (25). Thus if we plot a set of curves x = const., y = const. for small and equal intervals of x and y they will, locally, form a rectangular Cartesian net in the (-plane and, referred to it, the equation

02cp,

§2.2

cp(x, y;

" b 11 - f(Z')f(z)1 ". x, y) = - 2n In f( z) _ f(z') = cp(x , y,

x, y)

(34)

02cp

+ oy2 = O.

(30)

(For a more rigorous treatment see, for example, [9].) Suppose next that f - 1(0 is chosen so that as ( wanders over the boundary and interior of the unit circle '(I = 1, z wanders over the boundary and interior of some closed curve C in the z-plane, and thatf(z) is free of singularities inside C. Let zo, defined by f(zo) = 0, be the point which corresponds with ( = O. Then the potential b cp(x, y) = - 2n In If(z)1

(31)

obtained by inserting eq. (29) into eq. (27) is harmonic by eq. (30), satisfies

cp(x, y) = 0

on C

by eq. (28) and near Zo takes the form tpt», y) =

-~ In Iz 2n

b zol - 2 In If'(zo)1 n

+

Ojz - zol

with the singularity appropriate to a dislocation at X o, Yo. Consequently eq. (31) is the potential for a screw dislocation at X o, Yo in a cylinder with stress-free boundary C. To treat the case where the dislocation is not at the point Zo provided by the chosen fez) we could replace eq. (27) by the potential function for a dislocation not at the centre of the unit circle in the (-plane. Equivalently we can make an intermediate transformation which maps the required point onto ( = O. The details may be found in discussions of conformal mapping (e.g. [9]). Here we shall simply quote, and then verify, the result.

vanishes on C. Also near z' we have ip

=

-~ In Iz 2n

z'l

+ ~ iP(x', y') + 2n

0lz - z'l

(35)

with

1 - If(z')1 2 iP(x, y) = In If'(z')1 ' "

(36)

so that there is the required logarithmic singularity. The mapping function fez) is not unique since we can require it to send any point in C into the centre of the unit circle, but this ambiguity does not affect eq. (34). We may take any contour cp = const. as the boundary of a stress-free hole excluding the centre of the dislocation. If we needed a hole of another shape we should have to allow fez) to have singularities inside the hole to pull its boundary into the required form. However, if we are only interested in the conventional small circular stress-free hole of radius ao no modification is needed, since according to eq. (35) the contours are nearly circular near enough to the singularity. In this case the elastic energy is E =

til

f (grad W)2 dx dy = til f (grad.e)? dx dy

or by Green's theorem E = ill

f on

ocp cp-dS

180

J. D. Eshelby

Ch.3

taken over both boundaries with the normals respectively inwards and outwards on the inner and outer boundaries. Since cp is zero on the latter and is given byeq. (35) with /z - Zl/ = a o on the former, we have

J1b2

E(x', y') = -lP(x', y') -

4n

b2 ~4 In a o·

n

(37)

If we are not interested in the absolute value of E, but only its variation with position, we may omit the term In a o, and the result then applies also to the singular case with ao = O. The image force on the dislocation is Ex,

a

r, = (ax"

0)

2

/lb oy' ~ lP(x, y). I

I

(38)

§2.2

Boundary problems

181

Evidently with the help of a table of conformal transformations [l1J or by ransacking textbooks of electrostatics and hydrodynamics we could write down the solutions for an indefinite number of special cases. The following are a few of them. For a screw dislocation at (~2' 0) in the cylinder r = a we must take fez) =z/a, z' ~ ~2' Equations (33) and (37) then give eqs. (11) and (14) at once, together with

cp=

b In ar 2 r1

in the notation of fig. 2a. For a screw dislocation in a plate [12J we may find the mapping function by a physical argument. Figure 4a shows an infinite vertical row of dislocations of alternating sign and spacing d along the y-axis with a positive one at the origin. Evidently

According to eq. (26) we should expect to be able to write

r., r,

~ [(a~"

a:)

b 2

2

to get rid of the dislocation in the hole. However, this result is still not quite the correct solution. The last term in eq. (84) becomes -iD(~l - ~2)r2/a2 and represents a uniform hydrostatic pressure in the matrix. To get rid of it while still leaving the hole stress-free we add - D( ~ 1 - ~ 2) times the expression (68). The energy is the expression (86) with 1 and 2 interchanged plus corrections from the added terms. It takes the simple form ;:2

(86)

•

To find the elastic field of a dislocation with Burgers vector (b, 0) at P 1 near the hole r ~ a we start by changing t into r ' in eq. (71). The function X(t-1) shares with X(t) the property of having zero value and derivative at t = 1. Hence X = -DyX(r 1 )

(87)

satisfies the boundary conditions of eq. (54) on C2 and it is, in fact, biharmonic. The term - Dy In r 1 represents a dislocation (b, 0) at P l' so that the expression (87) is the stress function for a dislocation near a stress-free circular hole r = a. However, because of the term + Dy In r2 there is a Burgers vector (- b, 0) associated with the hole, and eq. (87) actually represents a dislocation at P 1 near a dislocation of opposite sign with a large hollow core. We may cancel the dislocation in the hole by adding the stress function of eq. (64b). With stressless terms rejected the result is clearly just eq. (72) with the suffixes 1 and 2 interchanged, plus (64b), that is X

ibDa2/~i

-

in the energy. This gives

2

where the first term corresponds to a simple image at P 1 and the second to the nonlogarithmic terms in eq. (84), comprising a two-dimensional hydrostatic pressure together with additional terms singular at P l' which can be interpreted on the lines of eq. (74) as a centre of dilatation, a dislocation dipole, and a doublet of centres of dilatation. To find the elastic energy we replace ~ 1 by a2 /~2 and integrate with respect to ~2' fixing the additive constant as in eq. (75):

2

(89)

(85)

~1 '

':> 2

blra?/~i

-bD In ~1

which like its analogue eq. (73) satisfies the conditions of eq. (52) rather than ofeq. (54). In the last term, r2 may, of course, be replaced by rf or d. Equation (84) can be further manipulated so as to agree with the results ofLeibfried and Dietze (quoted by Seeger [26J) and Dundurs and Sendeckyj [27]. The image (climb) force, now equal to the product of b and the yy-component of the image stress at P2' is easily found to be F;

-

in the image force which is equivalent to a term

It may be manipulated into the form r2

191

rr 1 + la 1 2 ( -1 - -ri = -Dy [ Inr2 r 2 ~i r~

)J

.

(88)

E = ibD In ':> 1 ~ a

2

+ const.

(93)

':>1

By combining the stress functions (71) and (87) we can find the formula which replaces eq. (71) when the inner circle Co is also stress-free, and thus test the accuracy of eq. (75). If the combination is chosen so that the coefficient of In r is - Dy it will take the form X

=

Dy{AX(t)

+ (1 -

- Dy[1n t - iA(t

A)[ - X(t-1 )J} 2

-

1)

+ i(1 - A)(t- 2

-

I)J

-DyY(t)

say, with t = d~dri~2' 2

(94)

which satisfies eqs. (54) and leaves C2 stress-free for any A. On Co its value - Dy Y(to) is of the form of eq. (53) and its gradient

aX ay

- D Y(to) - Dy [.. Y'(t)

~J ay

to

J. D. Eshelby

192

Ch.3

will agree with the gradient of eq. (53) provided the second term, or simply y'(to), is zero. This gives A

= 1/(1 + t5)·

(If the value of X and the y-component of its gradient agree with eq. (53) there is no need to test the x-component.) To get the associated energy we can evaluate the integral in eq. (66) between limits corresponding to t = to, t = 1: E

= jb

[OXJt=l = oy t=to

1 ( 1 = 2 bD In t;

-

jb[Y(1) -

1 - t 5) 1 + t~ .

§2.3

Boundary problems

193

directed at right angles to their Burgers vectors, and also, in some cases, concentrated couples as well. As a specimen we present Dundurs and Sendeckyj's [27J stress function for a dislocation with Burgers vector (b, 0) at PI in a matrix with elastic constants Ill' VI near the cylindrical inclusion C2 with constants 1l2' V 2 . The stress function is

+

Y(to)J

jD 1(Q - P)(x

~2)((-)2 -

(96)

(-))

in the matrix and X

(95)

This agrees with eq. (75) for small ao' To find the field of an edge dislocation with a general Burgers vector b ; = b cos tp, b; = b sin cp the above results may be combined linearly, For example, the stress function for such a dislocation in a stress-free cylinder is cos cp times eq. (73) plus sin cp times eq. (84). The energy is cos? cp times eq. (75) plus sin? cp times eq. (86) since the image field due to b; does not exert any force on the component by of the Burgers vector, and conversely, Leibfried and Dietze, quoted by Seeger [26J, state that eq. (75) gives the energy for any orientation of the Burgers vector if ao is much less than a. This is correct if we are interested in the value of E, but it is not good enough if we are interested in its gradient; the non-logarithmic term in eq. (86) gives an additional (negative) term in the force of eq. (85) which is comparable in magnitude with the contribution from the logarithmic term. We next suppose that fig. 2a represents a circular cylindrical inclusion (region 2) of elastic constants 1l2' V2 in a matrix (region 1) with constants Ill' VI' and that an edge dislocation is introduced at PI or P 2' The various cases have been worked out by Dundurs and his associates (Dundurs and Mura [28J, dislocations in the matrix; Dundurs and Sendeckyj [27], dislocations in the inclusion; Dundurs [7J, general review of this and related work). List [29J has given a unified treatment using the method of Muskhelishvili. Aderogba [30J has given a formula which might be useful in this kind of problem. It gives the perturbation due to the introduction of a circular cylindrical inhomogeneity into any initial elastic field. In a sense there are now two elastic fields associated with each region, its true field and the image field, the field of the other region extrapolated into it. We might hope to find the solutions in the way which served for the screw dislocation, namely by taking a linear combination of all the image singularities already encountered in connexion with the stress-free cylinder and cylindrical hole and then adjusting the constants to give continuity of traction and displacement at the interface. To realize this programme it is actually necessary to introduce additional singularities in the form of concentrated forces coincident with the image dislocations and

= -D 1(1

-

jP - jQ)y In r 1 + jD 1 (P

- Q)[(x - ~1)(-)1

-

(~1

-

~2)(-)T]

in the inclusion, and the elastic energy is 1

1

E = 2bD 1 [ z:(P

+

Q) In

;.:2

Sl -

~i

a2

-

1 a 2J 2(3P - Q) ~i

+

const.

(97)

with

Q = 1l1 K 2 - 1l2 K l 112

+

1l1 K 2

°

2n(1 -

VI)

In eq. (96) the new terms are those in 0, 2 ; eqs. (7) and (8) have been used to make the other terms resemble eq. (88). From eqs. (62) and (55) it is clear that the term (x - ~2)(-)2 represents a force transverse to the x-axis combined with a dislocation whose stress function might have been absorbed into the first line. The term - x(-) represents an opposite force and dislocation at 0 and the term ~ 2 (-) a couple at 0 which compensates the moment of the two forces. If the Burgers vector changes to (0, b) the forces are directed along the x-axis and the couple is not needed; its place is actually taken by a centre of dilatation. For a plane interface Head [31J devised an original method which deserves mention, though we shall have to refer to the original paper for the rather lengthy details. In plane strain one of the Michell-Beltrami compatibility equations reads + 02p/OX 2 = 0 which, since P = Pxx + Pyy is harmonic, may be written as 2T V = 0 with

r»:

T = Pxx

+ txP

and so, if we take x = 0 for the interface, continuity ofpxx is equivalent to continuity of the harmonic function T. If we arrange that Pxv is continuous at one point of the interface it is enough to require continuity of oPxyioy, or equally well of oPxx/ox elsewhere on it, by one of the equilibrium equations. This quantity can be written as a

J. D. Eshelby

194

Ch.3

linear combination of oT/ox and op/ox, plus a term which vanishes on x = o. Similarly continuity of displacement may be replaced by continuity of ov/oy and 02 U / oy2 and these quantities may be written, respectively, as a linear combination of T and P and of oT/ox and op/ox, plus, in each case, a term which vanishes on x = O. The boundary conditions are thus reduced to the continuity of the four harmonic functions

T

(98)

I 1- v -T---P

(99)

2f.1

o

2f.1

ox (T

I

(100)

- 2P )

~ (~ T +

1 - 2v

ox 2f.1

4f.1

p)

(101)

across x = O. Head now forms the linear combination v 1 ) T -1 - -- P ( ::1.+2f.1 2f.1

(102)

Boundary problems

§2.3

195

The two terms may be dealt with separately. The electrostatic analogy suggests that when the elastic constants in the region x < 0 change eq. (105) should be replaced by

c

L + C'*

ri

C'**

y,

d

L, ,2

x > 0 x
0. They write

dzz, = du~

+

du~

+

du~

(114)

where du ~ is the field in an infinite medium, du~ is the displacement of a loop of opposite sign at the image point (0, 0, e), so that if

then

and du~ is the additional displacement required to complete the cancellation of tractions on X 3 = 0. A loop of area dS, Burgers vector b, and normal n i is specified by the tensor bini dS and its elastic field is the sum of the fields of a set of elementary loops of the type b 1 n 1 dS, b 1 n 2 dS and so on in which all but one of the tensor components vanish, and so it is enough to exhibit the following special cases in which the Burgers vector and normal are parallel to one or other of the unit vectors i 1 , i 2' i 3 of the coordinate system: (i) prismatic loop: b = bi 3 , du~

= Ke[Ai3(R~1),i3 - (x3R~1),i33]' = bi.,

(ii) prismatic loop: b

du~

n = i3 :

n = ij'

= 1 or 2:

= K{(Aij - 2vAi3)(R~1),i - BR,ijj + 2V(X3R~1),i3 - e(x3R~1)ijj

+

2Be 6i3(R~1),jj

(iii) shear loop: b = bi.;

(iv) shear loop: b

= bi 1 ,

+

D[x/R

n = i3

du~ = Ke[ -Ai3(R~ l),ij du~

j

(1I5a)

+

(x 3R-

n = i2

+

x3

+

e)~l],iJ,

or b = bi 3 , 1),ij3]'

n = i.,

-

e( x 3 R -

1) ,i 12

or b = bi«,

+

D [x 1 (R

+

j

n = X3

e)-

1] ,i 2 } ,

where Au = 2v + 2B 6ij , K = b dSj4n(1 - v),

B

( ),i' ( ),i 1 denote ajax l ' a2jax i ax 1 and so forth, and repeated suffixes are not to be summed. In (iii) Groves and Bacon's image field and the geometrical image field defined by eq. (110) are equal and opposite, and for (0, (ii) and (iv) they are identical. The appropriate choice of sign is responsible for the simplicity of eqs. (115). A general finite loop in the half-space can be dealt with by dividing its discontinuity surface, any conveniently chosen surface spanning the loop, into a network of infinitesimal loops each specified by the tensor bin j dS. This tensor, in turn, can be decomposed into nine tensors in each of which only a single element is not zero, so that its elastic field is the same as that of one of the nine elementary loops displayed in eqs. (115a-d). The field of the complete finite loop can be built up by summation and integration. There is some simplification for a finite plane prismatic loop parallel to the surface [56]. In this case we have j = 3, k = 3 in eq. (111) and it is not difficult to see that du': only involves derivatives of r - 1. Likewise only derivatives of R - 1 occur in the appropriate du~, eq. (lI5a), terms in (R + X 3 + e) -1 being absent. Consequently the integration reduces to finding the Newtonian potential

of a uniform disk bounded by the dislocation line or its image. In the general case one 1 also has to find the corresponding biharmonic (or direct) potential in which is replaced by in order to find uf and [57], and to get u~ one must evaluate Boussinesq's first logarithmic potential ([2] p. 192) as well; for a single attracting point it has the form of eq. (118) below, and is responsible for the appearance of (R + X 3 + e)- 1 in eq. (115). (The expression in eq. (119) is Boussinesq's second logarithmic potential.) Bastecka [58] has calculated the elastic field of a circular prismatic loop with its plane parallel to the free surface. Division into elementary loops is not always the most convenient way of dealing with dislocations in a half-space. In particular, straight dislocations can be handled more directly. We begin with a screw dislocation running normal to a free surface [59]. In cylindrical polar coordinates r, e, z the field of a screw dislocation along the z-axis of an infinite medium is

Ir - r'l-

u:

Ir - r'l

00

Uz

b

e

= 2n '

pb 1

pz(J

t..

= 2(1 - v), D = B(1 2v)(1 - 26 i3 ) , 2 R = x~ + x~ + (x 3 + e)2,

(116)

= 2n --;:;

i

1),12

+

201

= 1 or 2: (1I5c)

= K{B[6 i1(R- 1),2 + 6i2(R - 1) ,1 + 2e 6i3(R- R,i 12

(1I5b)

Boundary problems

§3.1

(1I5d)

the other components are zero. There is a couple uba' about the z-axis on a circle of radius a in z = 0. This suggests that the image field for the semi-infinite solid z > may be constructed by introducing a distribution of couples along the negative z-axis. A couple at the origin gives the displacement ([2] p. 187)

u

= - curl (0, 0,

I Ot 2/b for a dislocation of Burgers vector b at the centre of a circular disk of radius R and thickness t. 3.3. Other three-dimensional solutions In this section we refer to some three-dimensional results which do not fall under the head of sects. 3.1. or 3.2. Salamon and Dundurs [56J have calculated the field of an elementary loop in a pair of bonded half-spaces, that is, an infinite solid with elastic constants J11' v1 for z > 0 and J12' V 2 for z < O. The method of solution is similar to Groves and Bacon's [55J (see sect. 3.1.) for the stress-free half-space, and starts from known expressions [67, 68J for the field of a point force in such a composite solid. Hsieh and Dundurs

209

Boundary problems

§3.3

[69J have considered continuous distributions of dislocations in the same composite solid. As an example they work out the field of a screw dislocation along the z-axis. With the notation of eq. (120) and the K of eq. (19) the displacement is u()

b r 2n R + r

z >

0 where

J1

= J11

b r for 2n R - r

z < 0 where

J1

= J12

= -K - - - for -K---

u = z

b 2n

e

in both regions. Some work has been done on the interaction of dislocations with spherical boundaries. Nabarro [70J gave the elastic field of an infinitesimal shear loop at the centre ofa sphere with a free surface, and Coulomb [71] found an approximate solution for a spherical cavity embraced by a concentric circular shear loop. Weeks et al. [72] found the elastic field for a straight screw dislocation threading a spherical cavity along a diameter. The change in energy on bringing the dislocation from infinity to this position is

where a is the radius of the sphere and ao is the usual core radius. They also gave the interaction energy of an edge or screw dislocation with a general spherical inhomogeneity in the limit where the ratio of the radius of the inclusion to its distance from the dislocation is small (see also [73]). Willis and Bullough [74] have calculated the field of a circular prismatic loop lying outside a spherical cavity, with its axis passing through the centre of the sphere, and worked out the attractive force between loop and cavity. Willis et al. [75J treated the case of a screw dislocation near a spherical cavity. They give a formula for the variation of the interaction energy with distance and study in detail the dilatation (which controls the motion of point defects) induced by the sphere in the otherwise dilatationless stress field of the dislocation. In these last two papers the problem was solved with the help of a series expansion about the centre of the sphere of the type presented by Love [2J in ch. XI of his book. The work of Collins [76-78] and Blokh [79] might perhaps be exploited in this kind of problem. They have given a general method for finding the change of elastic field when either a spherical cavity or rigid inclusion is introduced into a given field (see also [80J). The perturbed field can be written down at once as an integral involving the original field, but in evaluating it one may, of course, be reduced to using a series expansion. If the differences between the elastic constants of matrix and inclusion are small we may use eq. (108) to find the interaction energy with an inhomogeneity of

Ch.3

J. D. Eshelby

210

any shape. The energy given by eq. (108) may be written in the form

=

E

J

[W(/12' v2) - W(/1, , v,)] dv,

+

v

cos 28)

~~

(132)

.

(133)

To find its interaction with an inhomogeneous sphere of radius a centred at (R, EY), in the plane z = 0 we have to work out the integrals of 1/r 2 and cos 2{}/r2 over the sphere. The second is easy; since cos 2{}/r2 is harmonic its volume average over the sphere is equal to the value at the centre, and so we have at once

J

Cos 28 dv = 4n a 3 cos 2EY . r2 3 R2

=

+

p'2

+

2R p ' cos 12 -

!n - e12 ·

=

= (1 +

O:KL

2 A(K»(1

+

A(L»

E

(105)

K L·

+ Ii - 3h

2EK L ·

(106)

We may similarly obtain expressions for extensions and change of angles in deformed state coordinates by using a procedure similar to the above. For instance if we define the elongation A as the change per unit final length Idrl of the arc element so that

A = (ds - dS)

(107)

ds

239

tr f2]

K LP s .1 EM EN EQ 6e MNQ K L r-

(112)

From Vieta's theorem, however, it follows that IE

=

E 1 +

lIE = E 2E3

E 2 +

E 3,

+ E 3E1 + E 1E2 ·

IIIE = E 1 E 2 E 3 ·

For the linear theory we have simply O:KL :::::;

--

(104)

If we select the directions of dR 1 and dR 2 along the base vectors iK we get the simple relation

(113)

It can be easily shown that the roots Ea(o: = 1, 2, 3) are necessarily real and the directions N, associated with them are mutually orthogonal because E K L and H K L are symmetric in K and L. It is an interesting fact that the tensor f. itselfalways satisfies anequation analogous to eq. (Ill), namely _f.3

+ I Ef. 2

-

lIEf

+ IIIE' =

0,

(114)

which is known as the Cayley-Hamilton equation. It can easily be proved by using the identity det (I. f - f.. I) = 0 (see e.g. Lagally [37J). Uponmultiplyingeq. (114) by E - 1 we get another useful form of the Cayley-Hamilton equation

we obtain A(k)

Nonlinear elastic problems

§2.4

= 1 - (1 - 2ekk)

(no sum),

1/2

(108)

f 2

= IE f.

lIE'

g(kk)

where ekk are associated with elongations of the arc element originally parallel to the base vectors H k whereas the components ekl for k =1= I measure the corresponding shear deformations.

+ IIlEf. -1.

(115)

It follows that any power En can be expressed as a linear combination of " f., e:' with coefficients depending on IE' lIE and IIIE only. When the principal directions N; are taken as the base vectors of a suitable coordinate system the quadratic form ds 2 - dS 2 = 2EK LdX K dX L reduces, to the canonical form because in this case we have

2.4. Principal axes of strain and strain invariants Let us put eq. (27) in the form ds 2 - dS 2 dR dR = IdRI·f·ldRI = N·f·N, 2dS2

(116)

and from eq. (110) we get (109)

E\

=

(117)

E/jK a .

Hence we obtain where N specifies the direction of the vector dR in the natural state. Principal directions are those directions for which the above expression takes on extreme values. As is well known these are determined by the equations (f. - EI)·N = O.

(110)

This equation has three nontrivial solutions N 1 , N 2 , N 3 corresponding to the roots E 1 , E 2 , E 3 of the cubic det (f. - EI) = - E 3

+ I EE2

-

IIEE

+ IIlE = O.

The coefficients in this cubic are the invariants

h = tr f. = Ih =

!Ui

E

K K

=

~ tr f.2]

!e

=

K LP

!e

eMLpEMK,

KLP

eMNpEMKEN L'

(111)

E 1 1 = E1 ,

(118)

and therefore ds 2 - dS 2 = 2EK L dX K dX L = 2

L E a (dX a)2.

It follows from eq. (94) that the elongations A(a)

= (1 + 2Ea ) 1/2

-

A(a)

(119)

along the principal directions are

1.

On the other hand if we consider the quadratic form ds 2 - dS 2 2ds 2 = n- e-n

(120)

(12l)

Ch.4

B. K. D. Gairola

240

with

11

= 1 - (1 - 2ea ) 1/2 ,

+ lee 2

-

flee

and

+ Ille = O.

J = det [I

(123)

E

a e = --=-a 1 + 2E a

(124)

dS =

(125)

where (126)

As E. and e are related to Hand h through the relations of eqs. (29) and (30) we can easily obtain the following relations between the invariants of E., e, H, b, H -1 and h- 1 :

+ 2/E + 411E + 8111E =

(127.2) J 2,

(127.3)

= I h = 3 - 21e ,

(128.1)

IIH - 1 = Ll; = 3 - 4/e + 411e, IIIH - 1 = Illh = 1 - 21e + 411e - 8111e = J-

t, =

J- 1I H

;

2

•

(128.3)

(129)

These formulae were derived in detail by Murnaghan [35J. Apparently they were first given by Almansi [38J and Hamel [39]. The quantity J is nothing other than the counterpart of Jacobian} in the curvilinear coordinates. This can be seen as follows. From eq. (127.3) we have J2

= IIIH = det (H K L ) = det (G KM H M L ) .

dR(1) dR(2) -

(134)

dR(2) dR(1)'

(130)

dS =

(G)f JK L dx(1) dxa)

dS- =

(dX(1) dX(2) -

dS =

k

J

GL

= dS L G L,

(135)

= dS- JK GJGK·

K J dX(1) dX(2»GJGK

(136)

-! d 5 .. e

dS L --

or

l..(G)f

2

JKL

dS-JK

(1~7)

and dS = £·dS

(138)

or

where e is defined by eq. (A. 32). The two quantities dS and dS are called duals of each other. This procedure is quite general and can be applied to antisymmetric tensors of any rank. Similarly the area element spanned by dr(1) and dr(2) in the deformed state can be expressed in two ways, viz.

(128.2)

Furthermore, we get 2

(133)

dR(1) X dR(2)

It is not difficult to see that

(127.1)

2/E ,

Ilh - 1 = IIH = 3 + 4/E + 411E,

1

(132)

In view of eq. (48) and the relation (A.20) we can write eqs. (133) and (134) in the form

Ill, = J- 2 11I E,

IH-

+ IUVR + IluVR + IlluVR'

1

or as an antisymmetric tensor dS =

Il, = J- 2 (II E + 6111E) ,

Illh - 1 = IIIH = I

+ uVRJ =

The formula relating to infinitesimal area element in the natural state and the area element in the deformed state will be useful on several occasions later on. Let us consider the area element which is spanned by the two edges dR(1) and dR(2) of a parallelogram in the natural state. It can be represented either as an axial vector

I, = J- 2 ( / E + 411E + l211IE) ,

+

(131)

2.5. Deformation of area and volume

Since the invariants IE' lIE and IIIE are related to E a through eq. (113) and the invariants Ie' lIe' Ill, to ea through similar equations the formulae of eq. (124) permit us to express one set of invariants in terms of the other. For example, one can easily show that

I h- 1 = IH = 3

= [det (G KM A kMA kL ) ] 1/2 = det (A).

In view of eq. (33) we can also write

From eqs. (120) and (122) we get

ea E; = 1 - 2ea

J

(122)

where ea are the three roots of the equation det (e - el) = - e 3

241

Substituting eq. (70) in the above we obtain

= dr/ldrl and proceed in exactly the same way as above we obtain

A(a)

Nonlinear elastic problems

§2,4

ds =

dr(1)

d'S =

dr(1) dr(2) -

x

(139)

dr(2)

and (140)

dr(2) dr(1)'

Using eq. (68) and the relation H J

x

H K

= (H)fJKLH L,

(141)

eqs. (139) and (140) can be written as ds

=

d'S =

(H)fJKL dx(1) dxa)HL (dxt1) dxa) -

dx~)

=

dsLH

L,

dx(2»HJHK

(142)

=

d:sJKHJHK.

(143)

242

B. K. D. Gairola

Ch.4

The two forms are related by the equations ds =

±ill ..

or

€

ds = e -ds

dSL = ds J K =

or

±(H)f

(H)fJKL

JK L

(144)

ds L'

(145)

ds = ds L G L. A-I.

(146)

From H

l /2

L

CJKL

H l /2 l 2 = __ G / r G 1/2

CJKL -

H l /2 G 1/2

(G) f JK L

(147)

and eqs. (135) and (142) we obtain dSL = (H/G)1 /2 dSL'

(148)

Moreover, from eq. (127) we have H/G

= det (G K M H M L ) = det (H K L ) = IIIH =

J2.

(149)

Hence eq. (146) can be put in the form ds

= JdS L G L. A-I = J dS. A-I.

(150)

An equivalent formula was obtained by Nanson [40]. It is quite simple to obtain the formula relating the volume element d V in the natural state and the volume element dv in the deformed state. We have (151) and (152) It follows at once that dv

= J dV.

(153)

Similar relations were derived by Euler [41J and Cauchy [42J. The meaning of eq. (20) now becomes clear. It implies that no region of finite volume is deformed into one of zero or infinite volume. The conservation of mass is expressed by the equation Po dV = P dv

(154)

where Po and p denote densities in the natural and the deformed states, respectively. Substitution of eq. (153) in eq. (154) leads to another useful formula Po/p = J.

243

2.6. Compatibility conditions ds J K ,

In order to calculate the area change we can use either of the two forms. For instance, substitution of eq. (75) in eq. (142) yields

(H)r CJKL -

Nonlinear elastic problems

§2.6

(155)

If the displacement field components are not chosen as the basic dependent variables the question how to integrate the partial differential equations (35) and (36) [and equivalent equations (64) and (65) in curvilinear coordinatesJ naturally arises. Inasmuch as there are six equations for three unknown displacement components we do not obtain a unique solution in general if the strains are arbitrarily assigned. A unique solution may exist only if certain restrictions are imposed upon the strain functions. The equations that the strain components must satisfy in order that the displacements [or equivalently the mappings of eqs. (1) and (2)J be single valued and continuous are called compatibility equations. A physical meaning of the compatibility of strains can be understood in the following way. Imagine the body to be cut into small volume elements. If strains are assigned to each volume element arbitrarily (i.e. not satisfying the compatibility equations) and the volume elements are observed in the deformed state or vice versa, then it will be noticed that they do not fit together properly. There will be holes or gaps among them. On the other hand, if the prescribed strains satisfy the compatibility equations, then they will fit together properly with no holes or gaps among them. One can obtain the compatibility equations in a straightforward but rather tedious way by eliminating the displacement components from six equations. We shall, however, approach the problem from the geometric point of view which is more illuminating. In subsects. 2.1 and 2.2 we have implicitly assumed that the space in which the body is embedded is Euclidean. The Euclidean space is characterized by the existence of a global Cartesian frame, i.e. we can construct a Cartesian coordinate system over the whole space. In this global Cartesian system the components of a vector remain unchanged on parallel transport. Hence in a Euclidean space parallel translation of a vector is a unique operation, so that in making a circuit of a closed contour a vector does not change. Of course, a global curvilinear system of coordinates also exists in a Euclidean space. However, their existence alone does not guarantee that the space is Euclidean. To fulfill this requirement we must demand that the change in the components of a vector after parallel displacement around any closed contour be zero. Let us consider a body which is compact in the natural state. Clearly a single system of embedded Cartesian coordinates will cover the whole of it. The material space in the natural state is thus a finite region of the Euclidean space. Therefore, the embedded curvilinear coordinates X K which we used in subsect. 2.2 should satisfy two criteria in the natural state. The first criterion is that they should be global or holonomic. If the coordinate system X K is global the position vector R of a material point P is a continuous and single-valued function of the coordinates. This means dR = GK dX K is an exact differential and hence integral S~ dR = S~ GK dX K is path independent or equivalently

f ~f dR

GK dX

K

~ 0,

(156)

244

B. K. D. Gairola

Ch.4

where the integration is taken over a closed contour. We can transform the line integral with the help of Stokes's theorem into an integral over a surface bounded by the given contour. Taking into consideration eq. (156) and the fact that the surface is arbitrary we get aGK _ aGJ axJ axK

= «G)r L

JK

_

(G)rL)G JK L

= 0

or

Nonlinear elastic problems

§2.6

whereas eq. (157) or (158) is the integrability condition of the equation dR = GK dX K.

It is instructive to note that eq. (163) can also be obtained using the following arguments. Consider the vector a 2ujaX J aX M. In Euclidean space this vector would not differ from a 2ujaX M aXJbecause in a global Cartesian system the differentiation ofa vector reduces to ordinary differentiation of its components. That is 2u

(157) Conversely if the condition of eq. (157) is satisfied the coordinate system X K is holonomic, i.e. the position vector R is a continuous and single-valued function of the coordinates. The second criterion is that the change in the components of a vector V after parallel transport around any closed contour should be zero. Using eq. (A.37) we can express this fact in the form

f

(Glr;K

K

V

(l/p)iiKLEKL - y and the Gibbs function ljJ = q> - TS.

by = -1 (J-KL bEKL.

where bS is the change in entropy. If y denotes the internal energy per unit mass of the body then

L

(269)

In the same way one can also define two more functions, namely the enthalpy (259)

bQ,

Therefore, we can regard T and E as independent variables and consider n in the form y! = y!(R, T, f).

where bU is the increment of internal energy and bQ the heat acquired by the body. Furthermore, for every reversible thermodynamic process the second law gives us

1]

(267)

P

where ii KL is given by eq. (226). From the first law of thermodynamics we have

sU =

The increment b1] of 1] is

by! = -1 (J-KL bEKL - S o'I'.

From eqs. (254) and (257) it follows that bA =

257

Nonlinear elastic problems

§2.9

(266)

or

a

oy

= pA·-.A of

T

.

(273)

If the strain energy density W is defined per unit volume of the undeformed body, i.e. W(f) = Poy(f), we can write the constitutive equations in the form given first by Boussinesq [53]

1 oW T a=-A·-·A J of

or

kl (J

1

oW

J

KL

= - oE

k

A KA

I L'

(274)

B. K. D. Gairola

258

Ch.4

It is now obvious from the relations of eqs. (221) or (223) that T = a WlaE or r KL = a wlaEKL

(275)

and

= aw. AT

I

or

of

:L,KI

= aw AIL'

(276)

OEKL

These forms of constitutive equations are usually attributed to Kelvin [54J and Cosserat [55J, respectively. Ifwe express Was a function of A, which is possible since f is determined by it, we have aw oA kM

oW OEKL

Substituting OEKL oAkM

0 ( Am An ="2I oAkM 9mn K L -

G) KL

= 2"1 (bMK9km Am L + bML9kn An) K

1 CKLMNE E W -- 2! (2) KL MN

1 CKLMNPQE E E + 3! (3) KL MN PQ +

1 W = -2! (C (2) .. f) .. f

1

+ -3! [( C(3) ..

f) .. fJ .. f

I(OW

OW)

m

aw

m

= 2 ~ + ~ 9km A K = ~9kmA K' KM

(278)

MK

or

(279)

KM

(288) (280)

Substituting eq. (279) in the above relation we obtain 1 oW

(286.2)

or

k _ 1 oW m I (J'I - I aE 9km A KA L' KL

k

+

By definition the C coefficients which are called elastic constants are derivatives of W (see e.g. Thurston and Brugger [57J). For example

However, from eq. (274) we have

(J'I

(286.1)

or more compactly

in eq. (277) we obtain aw oAk M

259

Equations (281) and (285) are usually known as Neumann's [56J and Hamel's [39J forms of the constitutive equations, respectively. Ifwe consider a homogeneous elastic medium and suppose that WeE) is an analytic function of E, we can expand W in a Maclaurin series about R. When the initial state of the body is that corresponding to zero stress, i.e. the body is in the natural state, the expansion will begin with the second order term, so that

(277)

= OEKL aAkM'

Nonlinear elastic problems

§2.9

k

loW

or

= IaA I A K

(1

= --.

(281)

laA

k

From the symmetry of E KL and from the above definitions it follows that the elastic constants C KLMN and CKLMNPQ are symmetric in the following pairs of indices, (K, L), (M, N), (P, Q) and (KL, MN), (KL, PQ), (MN, PQ). Thus the maximum numbers of independent second and third order constants are 21 and 56, respectively. We now easily obtain (289.1)

On the other hand from (282)

or

and A 1K bA- 1Km = -A- 1KmbAlK

(283)

it follows that oW OAIK

oW oA- 1Mm oA -lMk OAIK

(284)

Hence we get k (J I

1

oW

= - J oA -1M A k

-1M I

or

(1=

1 oW

-laA- 1 · ( A

-1 T

).

(285)

oW -aE KL

= CKLMNE (2) MN + 2"1 CKLMNPQE -'(3) MN EPQ + ....

(289.2)

Substitution of eq. (289) in eqs. (274)-(276) yields series expansions for (1, T and I. The presence of material symmetry imposes certain conditions on the elastic constants. Because of the symmetry the elastic properties become identical in certain directions, and therefore some of these constants may be equal or otherwise simply related. Hence there remains the task of specifying the way the energy density is expressed in terms of the strains for a particular material symmetry. This will in subsect. 2.10.

260

B. K. D. Gairola

Ch.4

2.10. Material symmetry restrictions on the strain energy function and the elastic constants The symmetry properties of the undeformed material are characterized by a certain group of transformations which carry the crystal into a configuration which is indistinguishable from the original configuration. Therefore, the form of the strain energy function W must be unchanged by this group of transformations. If S is a symmetry transformation of the material we then have WeE:) = W(5 T f5). (290) Thus the form of W is limited in such a way that it can be expressed in terms of functions of strain which are invariant under the required group of transformations. Since we have assumed that W is expressible as a polynomial in the strain we can use a classical result of the theory of invariants (Weyl [58]) which can be stated as follows: Every polynomial scalar function of any number of tensors satisfying eq. (290) can be expressed as a polynomial in a finite number of scalars 11' 12 " •• , I k which are themselves polynomials of the tensors and satisfy eq. (290). None of these scalars are expressible as polynomials in the remaining ones. Such a set of polynomial scalars is called an irreducible integrity basis. These polynomial scalars can be determined easily with the help of five theorems on invariants given by Weyl (see also Green and Adkins [14]). In the present case each element In of an integrity basis is a polynomial in E satisfying eq. (290). For the isotropic case, for instance, the symmetry group Y of the transformations of the material coordinates is the full orthogonal group, and an irreducible integrity basis has three elements IE , lIE and IIIE. They are polynomials of the first, second and third degree in f. Therefore W can depend on E only through these three invariants, i.e. W = W(lE' lIE.' IIIE)· (291) The integrity basis for invariant functions of E for each of the crystallographic point groups has been determined by Smith and Rivlin [59J. That these are, in fact, irreducible was shown by Smith [60]. For the convenience of the reader we reproduce their results in table 1 at the end of this chapter. In this table E KL denote the Cartesian components because the symmetry in each case is conveniently described with respect to a particular rectangular Cartesian coordinate system which relates to the preferred directions in the material. This table also includes the transformations and their products which enter into the description of the symmetry for all crystal classes. The symbols denoting these transformations have the following meaning: (1) t is the identity transformation;

(2) (3) (4) (5)

N is the central inversion; Rl'R 2 , R 3 are the reflections in the Z2Z3' Z3Z1' Z1Z2 planes, respectively; ° 1 , °2 , °3 are the rotations through 180°about the Z1' Z2' Z3 axes, respectively; T 1 is a rotation through 90° about the Z1 axis followed by a reflection in the Z3Z1 plane with T2 and T 3 analogously defined; (6). M 1 , M 2 are rotations through 120°and - 120°,respectively, about an axis making acute angles with the axes Z1' Z2' Z3; (7) 51 and 52 are rotations through 120° and -120°, respectively, about the Z3 axis.

Nonlinear elastic problems

§2.10

261

The matrices representing these transformations are

I

[~

=

n

0 1 0

[-1

R 1 =.

, D =

,

[~

0 -1 0

H ~J

[~

s, =

°2 =

0

[~

0 -1

[-1

~

1

1 0 0

~ [~

0 0 1

1)3

1

51 =

-2

52 =

0

[1J~

R3

=

°3 =

-1

0

[~

M,

~l

0

1

T, =

H

0

0 1 0

0 0

~J

0 -1

~

-1

n ~ [~ n [-1J~ n

, T =

M

0 ~ 1 0

N=

[-1

H ~J -1)3 1

-2

0

T, =

[~

~J

0 1 0

[-~

0 -1 0

[~

~]

1 0 0

~J

n

Let us now consider the elastic constants of second and third order only. Since polynomials of degree greater than three in the integrity basis do not contribute to these elastic tensors.we shall form the strain energy function Was polynomials of second and third degree in E by choosing appropriate elements from table 1. This choice will depend upon the symmetry. In the case of isotropic rnaterials, for example, the second order constants are obtained from terms of Ii and lIE and the third order constants from Ii., IEIIEand IIIEin the strain energy. Therefore' using the relations oIE/iJEKL = b KL ,

(292)

oIIE/oE KL = IE bKL - E KL,

(293)

oIIIE/oEK L = E KM bKM + lIE bKL - IEEKL

(294)

we obtain 2

Ci/iMN = [

0 W ] = ).,a<jIMN OEKLOEMN IEj=O

+

J1dilMN

(295)

262

B. K. D. Gairola

Ch.4

and (3)

-

CKLMNPQ -

+

-

- v l b(l) KLMNPQ

263

and third order for each crystal class and their actual scheme has been worked out by several authors [61-64]. The number of independent fourth order constants for each class as well as the scheme of these constants for cubic and isotropic materials was given by Krishnamurty [65].

03W ] OEK L 8EM N oEp Q IEI=o

[

Nonlinear elastic problems

§2.l0

b(2) V 2 KLMNPQ

+

V3

b(3) KLMNPQ'

(296)

3. Nonlinear elastic problems in dislocation theory

where a~lMN =

bKL

+

b M N,

(297)

3.1. Discrete dislocations in an elastic continuum

+

(298)

The dislocation theory is based on the concept of the Burgers vector which may also be called the dislocation-displacement vector. In solid state physics the Burgers vector of a dislocation is defined in two ways. According to the procedure suggested by Frank [66J we first form in the dislocated crystal a closed circuit enclosing the dislocation line in a right-handed screw sense. In fig. 4a this circuit enclosing an edge dislocation is drawn anticlockwise and hence the positive sense of the unit vector I tangent to the dislocation line is away from the paper. We now draw the corresponding circuit in the perfect crystal following precisely the same sequence of lattice vectors, as shown in fig. 4b. The starting point S and the end point E of the circuit are no longer

aiflMN

=

b KM b L N

b}flMNPQ

=

b K L b M N b pQ,

b~tMNPQ =

bKL(b M P b NQ

+ + =

b

(328)

and

+

°

VR , (C(2) · · f (2») =

N,(C(2)"

- {-Tt

E(2») =

2)

-T*

(2)

on S S on c

= 0.

(329) Any solution of the above problem must satisfy the conditions that the body forces and the surface tractions are equilibrated in the deformed state. These conditions are given by eqs. (227) and (228). Using eqs. (154) and (212) these conditions can be put in the equivalent form

In view of eq. (319) these systems take the form

(U(n»)c

S,

B(l)

(U(2»)c

and (u(n»)c

= -

VR' l(n) = 0,

=

{oT onon S (1)

h.2)

+

VR' (C(2) .. f(l») = 0,

N· (C(2) .. E(l») =

(323)

By substituting the expansions eqs. (318) and (323) in eqs. (244), (313) and (307) we obtain the following successive systems of differential equations and associated boundary conditions: f(n)PO

+

n=l

n=l

n=l

en B(n)'

269

For n = 1 eq. (325) reduces to the boundary-value problem of tractions in the classical infinitesimal theory of elasticity. In general, when we have found a solution of the boundary-value problem for the (n - l)th step, we substitute it into the equations for the nth step which then have the same form as the equations defining the traction boundary-valueproblem ofclassical infinitesimal theory for the same material and same boundary. The body force and surface traction depend in an explicitly known way upon the previously determined fields u(1)' u(2)" .. , u(n-l)' Thus formally we are required to solve n boundary-value problems of traction in the classical infinitesimal theory for the same body. A disadvantage of this method is that the complexity increases very rapidly with the order of approximation and, therefore, it is not often practicable to proceed beyond the second order approximation which is suitable for moderately large deformation. However, at present only the elastic constants up to third order are reliably known and hence this is not a serious disadvantage. In most cases the scheme of Signorini which makes an assumption simpler than eq. (323) is more convenient. In this scheme one assumes that the body forces, surface tractions, and B are proportional to the parameter s, i.e.f(n)' T(n) and B(n) are set equal to zero if n ~ 2. In this case the system of equations up to second order approximation are given by

CX)

CX)

CX)

Nonlinear elastic problems

§3.2

(326) (327)

Iv f(l)PO dV + J dS 0, Iv r(R) x f(l)PO dV + Is r(R) X T(l)

(330)

=

T(l)

dS = O.

(331)

If we now assume that f and T are functions of R then the condition of eq. (330) becomes a necessary condition for the existence of a solution. The condition of eq. (331), however, cannot be applied directly because the position vector R is transrerrea

270

B. K. D. Gairola

Ch.4

Nonlinear elastic problems

§3.2

to r after the deformation, which may lead to a nonvanishing total torque. However, from a theorem ofDa Silva [79J it follows that the total torque of any system of loads acting on a body can be made to vanish by subjecting the body to a suitable rigid rotation. From this fact Signorini derived the following theorem of existence and uniqueness: If the external loads do not possess an axis of equilibrium and if solutions exist for the traction boundary-value problem in the infinitesimal theory then there exist solutions of the system (eq. (328». If there is an axis of equilibrium then there are also infinitely many possible rotations. However, uniqueness continues to hold if we prescribe rotations about each axis of equilibrium. The term axis of equilibrium denotes that axis around which the resultant moment due to a system of forces, whose resultant is zero, vanishes.

3.2.1.2. Example (i) Let us consider the simple example of an infinitely long and straight screw dislocation which may be produced from a perfect cylinder by shear displacement along the axis. To keep matters simple we shall restrict our attention to isotropic materials and the displacement field around the dislocation will be calculated up to second order only. This problem was treated by Seeger and Mann [18J under the assumption of vanishing tractions on the core boundary. We use cylindrical coordinates r, ip, Z, and let the body occupy the region rc < r < re where r c is the core radius and r, is the radius of the outer cylindrical surface (see fig. 7). In the natural state the coordinates will be R, ¢, Z. We shall use physical components as explained in appendix A.3. We first consider the boundary-value problem as one of anti-plane strain in which the surfaces r = r c and r = r, are traction-free and the displacement u suffers a discontinuity - B G, around any circuit enclosing the axis of the cylinder. We now use the method discussed in the previous section. Since we wish to obtain the stress and strain fields associated with a dislocation only, the body will be completely free of body forces. Therefore, in the first order approximation we have the system of equations

Fig. 7

271

Single screw dislocation in an infinite cylinder.

This is just the boundary-value problem of the linear theory and the corresponding solutions are well known. In cylindrical coordinates these solutions may be written in the form

=

U(1)

_ B(1)¢ G 2n z.

(335)

B(1) 4nR

E(1) = - - (G<jJG z

+

GzG<jJ),

(336)

J.1B(1) - 2nR (G<jJG z

+

GzG<jJ)'

(337)

I(1)

=

In the second order approximation we must solve the equations (338) (339) (340) Thus the displacement u is no longer discontinuous. For isotropic materials we use eqs. (302) and (321) and obtain C(2) ..

* _

I(2) -

(333)

= A(tr

1 [2..1 tr D(1)

+

I + 2J.1

1

-

2 V 1 (tr E(1»

(341)

£(2)'

2

+

-2

v 2 tr E(1)J I

J.1D(1)

(342)

Substitution of eqs. (336) and (337) in eq. (342) yields

* _ (A +

(334)

£(2»

+ 2j1U(1)VR ' £(1) + A(tr £(1»'V RU(1) + -2 + 2v2 (tr E(1) E(1) + 4V 3 £(1)'

(332) at at

£(2)

I(2) -

v2)B(1) 8n 2 R 2 I

+

(J.1

+ v 3)B(1) 4n 2 R 2 [GcfJ GcfJ

+

Gz Gz]'

(343)

B. K. D. Gairola

272

Ch.4

Nonlinear elastic problems

§3.2

(C

It can be easily seen that the boundary-value problem is now not only independent of Z but also of cP because of isotropy. Therefore we assume

- U(2)G R R'

(344)

U(2) -

Substituting eq. (344) in eq. (338) via eq. (341) and making use of the formulae in appendix A.3 we obtain V. R

I(2)

= G [(A R

2

2

+

2) (d Uk u dR2

~ d Uk

2

)

+R

dR

) _

Uk2 ») _ (A + /l +

V2

+

4n 2 R 3

R2

= 0,

v3 )

B(;)] (345)

1 d U~2)

dR 2

U~2) _ (A

+ RdR - R 2 -

+ /l + V 2 + v3 ) B(; ) 4(A + 2/l)n 2R 3

U(2) R

=

_{-GR'I(l)=-p~l)GR G R ·I (l ) = 0

(346)

2 + C2 R + A InR) B(1)' R

(CR1

(353)

NI . (1)-

This is the usual Euler differential equation. It can be solved either by the method of variation of parameters or more simply by introducing the independent variable t = In R which transforms it into a differential equation with constant coefficients. In any case, it can be verified that we obtain a solution of the form (347)

where (348)

The constants C 1 and C 2 are determined by the boundary conditions of eq. (339). In view ofeqs. (347), (343) and (339) we obtain two equations

A In R) (A + 2/l) ( C 2 - + - - AR2R2 R2

(352)

Thus the parameter 8 disappears in the final result. It is now a straightforward matter to calculate the strain and stress fields. (ii) Let us now consider the point of view that tractions on the inner boundary are non-zero. A simple assumption which is in accord with the symmetry of the problem is that the tractions on the inner boundary are approximated by a simple pressure P; . In accord with Signorini's simpler assumptions we shall put P; = 8 P(1)c' Thus in the first order approximation we must solve the equations V R ·I (l ) =0,

which leads to the equation d 2 U}/)

R) ]

1 cP In - G . -B [ -G +B - + CR+A 2n z R 2 R R

273

(u(1)\

= -

at at

R=rc R=re

(354) (355)

B(l) G z·

A solution of this boundary-value problem can be obtained by usual methods such as the Kolosoff-Muskhelishvili method [80]. We can, however, use the solution of a closely related problem, namely that of a stress distribution in a hollow cylinder under internal and external pressure, which was solved by Lame [81]. Consider a hollow cylinder infinitely long in the Z-direction, loaded by internal pressure Pc (pressure on the core boundary) and external pressure zero, with no body forces (see fig. 8) which corresponds to our case. The problem is then one of plane deformation with radial symmetry, independent of Z and cP. Therefore the displacement will have only the radial component. Thus, with the assumption of isotropy, the condition V R ·I (l ) = V R.(C(2)"

E.(1») = 0

(356)

reduces to

C1

C + A (R + C 1

2

2

+

R) + 8n2 A+Rv

In A R2

2 2

(357)

= 0

(349)

U~1)

for R = rc and R = r.: From these we get C 1 = A + V2 16/ln 2 C2

_

A [r; In rc

-

r~ In re

r; - r~

whence we obtain

_

A + 2/l], 2/l

= q1R +

(358)

Q2/ R .

(350)

_ _ /l_ ln r, A 2 2 A + u rc(re - rc)

-

(351)

From these results we can see that the displacement field can be written in the form U

=

8U(1)

= -

+

eB(I)

2 8 U(2)

[~ Gz + eB(l) (~l + C

2R

+A

l~R) G

R]

Fig. 8.

Hollow cylinder under internal pressure.

274

B. K. D. Gairola

Ch.4

The arbitrary constants q1 and q2 can be determined from the boundary conditions of eq. (354). We have

(1) -- 0 . E- R

E}Jl = q1 - q2/R 2,

(359)

It follows that (C(2) .. E(1»)RR = 2(A + J.i)q1 - 2J.iq2/R 2 , -

(C(2) .. E(1»)" = 2(A + J.i)q1

(360)

+ 2J.iq2/R 2 ,

(361)

(C(2) .. E(1))zz = 2AQ1'

(A

+ P;

+ J.i) Q1 - J.iQ2/ r; =

= 0

0

at

(362)

at

(363)

Hence

equation (346) now takes a more complicated form. After some straightforward but tedious calculations we find [82J

d 2U(2) 1 dU(2) U(2) 2AB 2 A 'p(1)2 R R R _ (1) c 2 2 dR +RdR- R -~+ R 5 '

+ J.i) (r; -

r~),

(A

I

A =

+ 3J.i + 2v2 + 4v3)r: y2 J.i2 (A + 2J.i) (1 _ y)2 .

C' U(2) = _1 R R

1 R

+ C'2 R + A!!.B2 + R (1)

+ 2J.i)(C

I

_

2

2 [

+ re2J • R

+

yp~1)

2J.i (1 - Y)

yp~1) [_J.i_ R 2J.i(1 - Y) A + J.i

R2

R

I

1

(366)

+ (Jc

+ nRJ, (367)

+ GzG ljJ 2' B k l and a", It can be easily verified that the additional quantities p* etc. are given by

283

Nonlinear elastic problems

§3.2

Putting x ' = r, x 2 =

qJ

and x 3 = z we obtain (see appendix)

(435)

1* + ~2 "" 11* + P* = ("" ~1 ,1,* = '1-'1

+ ~s "" 11* +

1* ~4 .t.

.t.

~3

111*)111 + ~ 111* 2111'

111* - J!....L 111* ~1 2111' .t.

+ ~6 "" 11* + ~2 "" 111*

,1,* = ~s "" 1* '1-'2

(428.1)

(428.2)

0

Gkl =

- 2111' ljJ 2 111*

(428.3) (429) (430)

B

2n

o

2n

g = G = r2,

2 a2 w O

As before denoting by 8

OJ 0 .

008

p'(8) = lim /312 = (1

oK

+ 8)/3

In view of eqs. (515) and (516) we obtain for the shear stress

Denoting by K the value of K'(e) when 8 = 0 we have

s:

[

8 t(1

(509)

oJi oV'

=

1 oJi "3 08

Ji the value of Ji(e) when 8 = 0 we get

I

=

"3 (3K +

P

+

3v 2

+

4v 3 ) ·

(519)

or Comparing this expression with that of eq. (491) we see that (510) where V' is the fractional increase of the volume of the material in the uniform dilatation. A comparison of eq. (510) with eq. (490) shows that

0K

C = -2-(1 +2-KoV' ) . K D

(511)

Next we consider an infinitesimal shear (512)

cS

1-OJi) = - -K1 ( 1 +poV' - .

(520)

We can, therefore, put eq. (495) in the elegant form 1 OK) e- = - -K1 [( 1 + K oV'

-

U;; D

1 0 Ji) J¥: - ] + (1 + PoV' s·

(521)

The procedure described here can be applied to materials of any symmetry. However, the computations are rather lengthy for materials of lower symmetry. We only

Ch.4

B. K. D. Gairola

294

mention here the Zener's formula for cubic crystals derived by Toupin and Rivlin [97] which can be put in the form

e-

1 [( 1 aK) = - K 1 + KaV' W D + ~~ + J1I aail) V' W s +

( 1 + a1 aaft) - 'J V' Ws ,

Nonlinear elastic problems

§3.4

Since our division is fine enough we may regard A -1 as a continuous function of position r, and continuously differentiable. We assume that the inverse relation dr = A·dR

(522)

V

2

azz azz

azz aZk

±(auaZaa)2 J

dV

a>

A= (523)

If the crystal contains a large number of dislocations it is convenient to regard their distribution as continuous and introduce a dislocation density which is a function of position. This may be done by using a limiting process in which the number of dislocation lines approaches infinity while the Burgers vector of each decreases in proportion so that the number of dislocations times their individual Burgers vectors remains finite in any bounded domain. We further assume that the number of dislocations times their individual core volumes tends to zero, i.e., we smear out the cores. From these considerations it follows that the natural state of a continuously dislocated body can be realized only by dividing it into infinitesimal volume elements and letting each element relax separately (by making appropriate cuts). The relaxed elements can be considered as tiny fragments of a Euclidean space and as such we can introduce in each of them a local (embedded) rectangular frame (K with the base vectors i K . These frames may be chosen in such a way that in every element they have the same orientation with respect to the axes of material symmetry or they may be identical to lattice lines so that i K are lattice vectors. The same natural state, of course, can also be realized by dividing an ideal crystal into infinitesimal volume elements and deforming each of them by the creation of slip planes which corresponds to the creation of dislocations and motions of dislocations. Such a procedure leaves the lattice lines unchanged and corresponds to the plastic distortion defined by Kroner and Rieder [98]. Therefore an ideal crystal structure exists in the elements in their natural state. It is clear that each element can translate and rotate freely in the natural state. We may, therefore, assume that each element is rotated in such a way that all local frames become parallel. In other words the local frames are just the fragments of the global Cartesian frame. The mapping which carries an element from the deformed to the natural state is defined by the relation or

d(K

= A -lKk dx",

(527)

;PKUkiK'

04.04- 1 = 04- 1.04 =

3.4. Continuous distribution of dislocations

A -l·dr

(526)

also exists, so that

1

with summation convention suspended for the Greek indices.

dR =

or

where

where a is the second shear modulus of cubic materials and

W~ = ~ fa [~ (aUk aUk + aUk au z) -

(524)

where

295

I.

(528)

The transformations of eqs. (524) and (526) are not the same as those given earlier in sect. 2 (such as those of eqs. (6) and (12» because A mid A -1 cannot be represented here as the gradients of a vector field. The reason is that after the process of relaxation the volume elements do not fit together to form a continuous body. Hence the above transformations are not integrable. It follows that the compatibility conditions discussed in subsect. 2.6 are not fulfilled. For this reason we use the term elastic distortion, as suggested by Kroner [24], for A and A -1 to distinguish them from the coefficients describing the previous case. We can, however, use all the formulae of sect. 2 involving the deformation tensors A and A -1 provided we replace them everywhere by the distortion tensors A and A- 1. We now generalize the definition of the true Burgers vector as given by eq. (306). The true Burgers vector of all dislocations enclosed by a small circuit in the deformed body is f).B

=

rl -

A -

1

(529)

. dr.

By the application of Stokes's theorem we get

/'..B

~ 1: be

+ [b(Jo)] .. e -

+

(594)

+ 2j1e(1)'

e

+ /1")1; - 2Jl"110

qJ(lo' 110, IlIa) (tA"

+ +

(596)

= Ale I + 2j1e(2) + (A1Ie~1) + A 2Ile(l)I (2)

A 3Ie(1)e(1)

+ A 4IIle(1) e(l} ,

(597)

A 1 = tv~

+ vi - A, A 2 = - 2(vi - 2v~ - 2j1), A 3 = 2(vi + 2v~ - A - 3j1), A 4 = 4(v~ - /1).

(598)

Usually the constitutive equation in the second order approximation is written as 8

2

0(2)

= AIe' + 2j1e + (A 11e2 + A 2IlJI + A 3lee + A 4Illee- 1, where we have put e = s e(l) remaining terms.

+

8

2

e(2) in the first two terms and e =

4v~IIla

+ ....

(604)

(599) 8

e(l) in the

(605)

Using formulae similar to eq. (593), i.e. 81a/80 = I etc., and proceeding as before we finally obtain

e(l) = It,'''10(1)

+ 2 j1 0(1)

(606)

II

e(2) = A"I a(2 ) + 2/1" 0(2) + (A~ 1;(1) + A~IIO(l)1

+

+

+ 6v~ + 8v~)I; - 2(v~ + 2v~)Iallo

... 8qJ I (8qJ ... 8J) e = 8(J0) = J 80 - 0" e 80 .

where

o = 80(1)

k(v~

Furthermore we get

By the application of the Cayley-Hamilton theorem (see e.g. eq. (115» we can put eq. (596) in the form

+

(602)

(603)

(595)

0(2) = Ale(2) I + 2j1e(2) + [(!v~ + v~ - A)le~l) - 2v~IIe(l)]1 + 2(v~ - A - j1)le(1) e(1) + 4(v~ - j1)e 2.

()(2)

bW

or

Substitution of eqs. (589), (594) and (128.3) in eq. (588) yields

0(1) = Ale(1) I

(600)

it follows that qJ can be considered as a function of 0 and we can expand it in a power series in terms of a. For isotropic materials we have

8W/8e = AIe' + 2j1e + (!v~ + v~)le21 2v~(Iee - lIe/)

18W '

J 8e

W.

bqJ = [b(Jo)] ..

we get

+

- 2e) =

where e is Hencky's logarithmic strain measure defined by eq. (32). We now introduce the Legendre transformation

U sing the relations

81e/8e = I,

305

We now determine the inverse relation which expresses e in terms of a. For this purpose we first rewrite eq. (588) as

u' = j1,

vi = 2(,1 -

Nonlinear elastic problems

A~lo(l) 0(1)

+ A~IIIo(1) 0(1:'

(607)

etc., where

=

3v~

A~

=

;j1 (A"

A~

=

~ (A" + 2j1")(3v 2j1

A~

=

v~ -

+

v~

+ (A" + 2j1")2(3v - 3v2 - 1),

A~

+ 2j1")(1 - v) +

1/4/12

2) -

v~, v~, (608)

B. K. D. Gairola

306

Ch.4

and v is the Poisson constant. Moreover, if we compare the first order equation (606) with the first order equation (595) we see at once that

A" = Ji"

2Ji(3A

+

4Ji

= o.

V,,'X

1 Ji = 4Ji'"

1, =-

Nonlinear elastic problems

(609)

307

From the existence of null tensors it follows that X can be made to satisfy various gauge conditions, e.g.

A"

A=

2Ji)'

§3.6

(618)

We now consider eq. (584) together with the equilibrium equation (614). We shall assume that (619)

Thus one set of the formulae is given by merely interchanging the symbols A and A" and Ji with u" in the other set. Similarly the relationship between v~, v;, v~ and v~ , v;, v; is given by the following equations

ex)

L

e =

e(n) ,

En

(620)

n=1 ex)

v~

!v~

+ tk3(9v{ + 18v~ + 8v~) = 8K, 12JiK2(3v~ + v~) = - 18K(2 - 3v)j(l +

+ 2v; + 3v; + v~ + v;

+

8Ji3V~

a

v),

= t Ji.

+

(610)

2Ji"0

+

(A~ l;

+

A~lla)'

+

+

A~llIao-1.

(611)

3.6.2. The method of Kroner et al. The starting point in this method is the fundamental geometric equation (584). Since the displacement function method is not applicable here this equation is first written in terms of the stress 0 and then solved by the stress-function method which makes use of the identities

V,,) = 0, + (V"a)T]

div Inc"X = V,,, (V" x X x Inc" Def a =

tv"

x [V"a

(612) x

V" =

O.

(613)

These may be considered as counterparts of the well known vector identities div curl a = 0 and curl grad! = O. In the absence of body forces the stress equation of equilibrium

V,,'O = 0

(V"a)T] ,

(616)

which plays the role of a null tensor because, in view of eq. (613),

= Inc"xo = O.

(622)

= O.

(623)

and hence h np = gnp

+ 2ejkgnje" = gnp + 2(8 ej~) + 82 eji) + ... )gnjgPk.

(624)

We now consider the case of isotropic materials. Inserting the relations of eqs. (606) and (607) in eqs. (622) and (623) and using eq. (608) we obtain

m

+ m + l(V" V" -

I !.1,,)Ia

(l )

(625.1)

= 2Ji1'/(1)'

V",, 0(1) = 0,

(625.2)

and V" 0(2)

m

+ m + (V" V" - 1!.1,,)la(2) =

Vr' 0(2) = 0,

2Ji(P(1)

+

Q(1»)'

(626.2) (626.2)

where !.1,. = (g)V/g)Vj is the Laplacian operatdr, m is the reciprocal of Poisson's constant, i.e. 1

(617)

V,,, 0(1) = 0,

The meaning of 1'/(1) and Q(1) = 8(1) + ~(1) is obvious. Note, however, that in the expressions of eqs. (201) and (583) h'" should be replaced by gnp, since according to eq. (65) h np = gnp - 2enp

(615)

+

1'/(1)'

V,,'0(2)

V"O (1)

where X is called the stress function tensor. It follows from the symmetry of 0 that X, too, is a symmetric tensor, i.e. X = XT. The solution of eq. (615) is also known as the Gwyther-Finzi solution [107, 108]. Finzi also noticed that the tensor X is indeterminate to within an arbitrary tensor

0°

=

(614)

= Inc"X,

X° = Def a = t[V"a

"e(1)

and

is identically satisfied by letting o

(621)

Substituting eqs. (619), (620) in eq. (584) and eq. (621) in eq. (614) and considering only terms up to second order we obtain the following system of equations: Inc

A~lao

En O(n) ,

n=1

In second order approximation we may also write

e = A"la'

L

=

2

m = - = - (A v A

+ Ji)

B. K. D. Gairola

308

§3.6

(627)

The fact that eq. (636) is an admissible condition can be demonstrated in the following way. If XU) is a solution of eq. (634) which does not satisfy the auxiliary condition of eq. (636), we can introduce another function XU) = X(1) + X?1) which does. The vector a in X?1) = def a is then determined by the equation

and

with (628)

When the right-hand sides ofeqs. (625.1) and (626.1) are zero they are called BeltramiMichell stress equations of compatibility [109, 110]. In deriving these equations we have used the fact that

Inc ,.« = Vr x (J

= ~r(J +

X

Vr

def(Vrla - 2V r'(J)

+

I(Vr'(J,Vr - ~rla)

-

~ra

(1 ) +

+ 2m+3 2 Vr v..« + 2Vr ' m +

is.u..

Since this equation possesses a non-trivial solution one can always find a solution = X(1) + def a which satisfies eq. (636). An alternative form of eq. (638) is (640)

(630)

o=

~r~rX(1)

and rewrite eq. (625.1) in the form

IL'.,) I""

=

~(')'

(632)

(633)

we obtain

+

M

=

(634)

11(1)'

The tensor M is given by

M=

def[(m :

21L'.}

I V,V, -

X(l):V, +

m ~ I V,L'.,Ix".}

(635)

Xl')

=

(636)

0

with X,(1)

= ~(X 2/1 (1)

-m+2

~r~r2/1( X(1)

+

_1-1 IIx' \

m

+

~(l) ~ L'.,L'., X(l) + m ~ I I If (~(') - L'.,L'., X(l))}

01 (641)

f ~(dr - r'l dv',

= -

gIn

~

g~ f ~(l)lr -

-

(642) (643)

r'l dv'

are, of course, fully equivalent. Similar relations were first given by Moriguti [101], but without using the stress function tensor. These solutions, together with eq. (633), furnish us with (J(1)' From the relation of eq. (606) we then obtain the corresponding e(1) and hence also Q(1) and P(l)' The second order equations (626) can now be solved in exactly the same way. One obtains

~

-

g~

f

(p(l) + Q(1))lr -

r'l dv',

(644)

which determines (J(2) through (J(2) = Inc r X ( 2) and e(2) through eq. (607). The final solutions, in the second order approximation, are then given by (645)

_I_II

),

X(l)

(637)

and eq. (634) reduces to an inhomogeneous biharmonic equation ~r~r X(1) = 11(1)'

X(l)

X(2)

However, it can be seen that this tensor vanishes if Vr ' X ( 1 )

21{

"(ly

Equations (638) and (640) and their solutions

(J(1) = Inc r X(1)

X(1)

11(1)

m - 1

Inserting in this equation

~r~r

-

= 2/1( 11(1) + _1_ it; \ =

I V,V, -

(639)

.

(1)

X(1)

(631)

(m:

- - 2 I I xm

which is obtained by inserting eqs. (631) and (637) in eq. (638) and taking the trace of the equation:

In order to simplify eqs. (625) and (626) we now use the Kroner-Marguerre [Ill, 112] procedure. We first introduce a modified function

V,u(l) +

X(1)

(629)

and Incr(lIa ) = (Vr Vr

309

Nonlinear elastic problems

Ch.4

(638)

The solutions given by eqs. (642) to (644) are valid for infinite media only. However, as Kroner [91J has shown, these solutions remain valid also for finite media provided 11 is replaced by a function Tj which may be considered as a continuation of 17 beyond the region occupied by the body and which vanishes rapidly enough at infinity.

B. K. D. Gairola

310

Ch.4

§3.6

Nonlinear elastic problems

311

So far the use of the stress function method in elasticity theory has been confined almost entirely to plane problems or torsion problems for prismatic bars. The dislocation problems which have been treated by Kroner et al. are also of this group. F or such problems a slight modification of the method described above proves to be convenient. For the sake of simplicity, let us use the Cartesian coordinates Zk and assume that the problem depends upon Zl and Z2 only, i.e. o( )/OZ3 = O. Then the first order fundamental equations are given by 02eW oal 1l OZ~

= - OZ2 '

02eW oa~ll ozi = OZl ' 2 0 eW _ aW oa~V») OZ10Z2 - -2 OZl - OZ2 '

~ (oeW _ oe~ll) = ~(oaW _ OZl

2 OZ2

AF) Ll

(648)

(1) .

(649)

(oe W oe~ll) _ 1(oa W oaW + oa~V) OZl OZ2 - OZl - 2 OZl - OZl OZ2'

o

OZ2

(1)

Substituting eq. (648) in eq. (646) and performing some trivial integrations we finally obtain the following first order equations

1(0

OZ2

1

(1) _

e33 - 2f.1(m + 1) m(j33 -

~ ( 1 -

§3.6

= 0,

a~) = 13(1)'

+

.. 13(1)

+

C(2)" (13(1)' 13(1)

.. 13(1)' f3T1)

- 1P( 1 ) C(l) .. 13(1)

+ t C(2 ) .. (f3~)' P(1) + t( C(3) ..

f3(1) .. f3(1)'

(702)

Thus, in the present case, we have to solve the following successive system of equations f3(1) x

VI' =

-(1(1)'

Vr · ( C (2) .. f3(1)

= 0

(703)

and (704)

B. K. D. Gairola

318

Ch.4

These equations should be supplemented by appropriate boundary conditions. In the following we shall consider only an infinite medium and so assume

101-+ 0

11'1-+

as

where

~ V~( I' + Vip

-aZj kl

tiji

_

a 2f3(2)

e(2)~ijkl a

Zj

a

ZI

CXki '

I'

')

= O.

(712)

a

C(2) a (f3(1) (1)) i jkl f mIq kp CXpq Zj L

i jklnp

(705.1)

(1)

-

Proceeding in exactly the same way we obtain the following equation from eq. (706):

+ af3(l)

319

1") is the well known elastic Green's function satisfying the equation

Ykp(r -

(2 ) a2YkP e ijkl ~a OZj ZI

O.

The higher order equations are all of the same type, i.e. linear with respect toP(n)' These equations may be solved in the usual way by means of Green's tensor functions. Let us use Cartesian coordinates Zk' Then the first order equations (703) may be written as

Nonlinear elastic problems

§3.6

~ a a (f3(1)f3(1)) kl np Zj

-;

-

0,

(713)

which has the solution

af3(l) e(2) _k_1 - 0 ijkl a -, Zj

(705.2)

(2 ) -

f3km -

-

faYkP(r aZ.

1") e(~)

p jql

f

lmn

f3(~)(r')cx\l)(r') d V' ql In

]

and the second order equations (704) as af3(2) t .. _k_1 !]I Zj

= -

a

(706.1 )

f3(l)CX(1) km mi > a(f3( 1 )f3( 1 ) )

2

af3( ) e(2) _k_1 +L ijkl

-

aZ.

ki

mn

0

-

oz,

ijklmn

]

(706.2)

-,

]

where LijkImn

=

6Im + eHJ 6mn

eU;;m

-

+ ±eg~ 6km + ±eSUmn'

If we multiply eq. (705.1) by af3(l) kn aZ m

s.; + ei~kl s;

eUiJn

af3(l) km _ aZ n

timn

aykp (I' -

f

aZ

1")

j

a

L.. [f3(l)(r l ) f 3 ( l ) ( r')] p]!lnq aZ~!1 nq

dV '

(714)

.

To this solution we can add any appropriate solution of the homogeneous equations corresponding to eq. (704). This solution would be of the same order as P(2)' In the case of a screw dislocation in an isotropic medium such a solution would then correspond to those in subsects. 3.2.1.2 and 3.6.2.1. In the present method, however, this is not done because, as pointed out in subsect. 3.2.1.1, the assumption of vanishing traction on the core boundary is, physically, not appropriate. It is interesting to note that if we use the true dislocation density and assume as before that

a

(707)

(715)

we get we get

(1)

(708)

timn CX ki

a(l)

= -

P(1) X

Vr =

(716)

a(1)

and

and hence

(717) (709) We now differentiate eq. (705.2) with respect to Zm and substitute eq. (709) in the resulting equation. We obtain the following equation: 2 f3(1) (2) a km C ij kl - a a Zj ZI

+

a (1) (2) CX kn _ CijkIfmIn-a- Zj

(710)

O.

(1 ) f3k3

For an infinite medium a particular solution of this equation is ( 1) f3km

= -

aZj

faYkP(r -

1") C(~)t pJ!1

which implies that P(2) is the gradient of a vector field. This means that in this case the first order equations are the same as before and the second order equations are just those of the elasticity theory which we have considered earlier. A somewhat similar situation arises if the only nonvanishing component of the dislocation density is CX 3 3. From eq. (711) it follows that in this case

=

faYkP(r a Zj

1") C(~) e

p]!1 3In

cxP)(r') in

dV ' = 0 ,

(718)

and hence, from eq. (706.1), mln

cxP)(r') In

dV '

,

(711)

(719)

B. K. D. Gairola

320

Ch.4

i.e,

(720) Thus here too the second order problem is the same as in the elasticity theory. It can be seen that the first term on the right hand side of eq. (714) also vanishes. Therefore, we have

f3k~

J8YkP~ -

= -

r')

Lpjilnq

[f31i)(r')f3~~)(r')J dV',

8

Zj

(721)

§3.6

Nonlinear elastic problems

321

which have a solution for Ai provided that the determinant of the coefficients vanishes. P This yields a sixth order equation in P having six complex roots denoted by P P(3) and complex conjugates p(l)' P(z), P(3) and corresponding to these roots t~~;e g)~ set A(Il)i which satisfies eq. (726). The quantities D(Il)j satisfy the equations Re

tt,

A(,),D('li}

0

=

(727)

and

and it follows from eq. (720) that U(2)

k

(728)

= _ J8 YkP(r8 - r') L.. R\!)(r')R(l)(r') dV' . p jilnq Pzl Pn«

(722)

Zj

The main difficulty in the above method lies in deriving the elastic Green's tensor for anisotropic materials. Until recently it was a formidable task. But now this obstacle has been overcome to a great extent as a result of the work of Indenbom and Orlov [113]. Their ingenious method relates the three-dimensional Green's tensors to the two-dimensional ones and these are easier to calculate. The underlying theory is, however, conceptually rather complex. In certain situations one can use the straightforward methods suggested by Willis [114J and Mura and Kinoshita [115J*. 3.6.3.1. Example. We consider the example of an infinitely long straight screw dislocation in an anisotropic medium. We assume the Z3 axis to be parallel to the dislocation line. In this case the dislocation density is given by eq. (667) and the state of stress depends upon Zl and Z2 only. We now write eq. (705) in the form

8 2 Ykp ) C i(cx2kP -8 8 Zcx

zp

5: "5:(Z1 + "i»

-

,

Zl, Z2 -

') -- 0 ,

(723)

Z2

where Greek indices take the values 1,2 only. The Green's tensor Ykp satisfying the above equation can be obtained from the solutions to the linear elasticity equations such as the one given by Eshelby, Read and Shockley [116]. Using their result Willis has shown that

yJz, -

2'"

Z2 --

z~) =

Re

t:

Pi)) In [z, - z',

+ P(,)(Z2

-

Z~)]},

It is wel~ known that the condition for the existence of a simple solution of pure screw type IS that the plane Z3 = 0 be a plane ofelastic symmetry. The elastic constants ~t a point have the same values for every pair of coordinate systems which are mirror Images of each other in this plane. This requirement is valid in the present case since we are considering a singular screw dislocation as a limit of a continuous distribution of straight screw dislocations. Consider now two coordinate systems Z: and z , with z~, z~ axes coinciding with the Z1' Z2 axes parallel to the plane of elastic symmetry. However, c~oose the z; axis with z~ = - Z3' so that one system is the mirror image of the other In the plane of elastic symmetry. Consideration of the tensor transformation or, more simply, consideration of the definition and sign conventions of the components shows that

and

(729)

while all other independent strain components are unchanged. Hence the limitation imposed on the form of W is W(E~l' E~2' E~3' E~2' E~3' E~l) =

W(E l l , E 2 2, E 3 3, E 12, - E2 3, - E3 1).

(730) It follows from the theorems on invariants that the integrity bases are E E E E 2 2 11' 22' 33' 12' E 2 3, E 3 1, and hence

(724)

(731) It can be easily verified that

where Re means "the real part of" and

L3cx3P3y

= O.

(732)

Equation (724) may now be written as The quantities Aln)i and p(ll) are defined in terms of elastic constants Ci7Jl by the three homogeneous equations (2 ) A i [C jlil

+

(Cl2)

jli2

+

C(2) ) j2il ifJ

+

c(2) 2 j2i2P

J = 0,

* For recent developments see Ch. 2 by Steeds and Willis.

(726)

Y.P(Z, - z;, Ycx3(Zl -

2, -

Z~,Z2

-

z~) = z~)

Re

Lt, P~p)

In [z, - z;

+ P(,)(Z2

-

z~)]

r

= 0, (733)

Ch.4

B. K. D. Gairola

322

Nonlinear elastic problems

§3.6

323

where Y(v) = ZI

(734)

+ P(v)Z2'

(742)

cx(v) = (P(3) - p(v»)j2jpi~

where

o, = bPg)t3ap(C~~)31

+ P(3)C~~)32)'

(735)

All other components of P(I) vanish. Therefore, from eq. (722) we obtain U;2 l(Zl' Z2) =

f f iJy«P(

-

Z

l - ; : ; Z2 -

z;) L PA3 " 3 )

x f3~IJ(Z~, z~ )f3W(z~ , z~) dz~ dz~,

ff

OY33(z 1

-

z~, Z2 - z~) L

:::l

oz ,

(736)

f3 (1)(Z' z') 3B/l3y 3/l l ' 2

and pi~ etc. denote the imaginary part of p(v) etc. The fact that II is discontinuous for = 0 does not make any difference here because using the fact that the strain energy must be positive Eshelby et al. [116J have shown that the roots P(n) are never real. The integral 12 is singular, which reflects the fact that the linear solution is singular on the dislocation line. However, as we are considering the singular dislocation as a limit of a continuous distribution of dislocations, we need consider only the finite part J 2 of the integral 12 which is given by 2 J = - npi0 [CX(V)!Y(v)1 - (l + CX(v»)ytv)] 2 lim! im n (1 + CX(v) *)1 y(v) !2 - CX(v)y(v) * 2 p(v) P(3)Y(v) pi~

{I

_ 1 (- 2) _ 1 1 n y(v) n 1

' ' (737) f3 (3yI )( ZI' Z2, ) dZI' dZ2· However, we can see that U~2) vanishes because of eq. (732). Hence we need consider only eq. (736). It follows from eqs. (733)-(735) that X

(2)

Ua

2

1

_

Re { 4

(z l ' Z2) -

X

»»,

D/lD~

+,

+

,

l U;2 (z l' Z2) = -in Re {

OO

f

00

foo _

CD

dz' dz'

(Z'l

+ P(3)Z~ )2[Z1

~ z~

2

p~~\ irn

+ p(v) (Z2 -

,tl

P;ply (,l(L fI1 3" 3'

+ PM L p2 3"3,)

(738)

D/l D y ( 1 1 ) 2 x [~ 2 2 + cx(v)p(v) cx(v) 1y(v) 1 - (l + cx(v») y(v) (l + cx(v») y(v)

The two types of integrals which occur here are

_

if

2

+ , D*D*] /l *Y, 2. dz'1 dz'2 } . (Z1 + P(3)Z2)

II =

p(v)

(l + cx(~»)IY(v)12 - CX(~)ytV)] n. CX(v) 1Y(V)12 - (l + CX(v») ytV)

(v)

D~Dy

jz1 + p(3l2!

{I [.

p~~\ > 0 1m'

if

< O. (744) p(v) The final solution will then approximate the true solution away from the Z3 axis. Substituting eqs. (741) and (744) in eq. (738) we get

. ,

x, , 2 (Z1 + P(3)Z2)

npi0

!pi0Ipi~)y(v)

+ CX(V)} + CX* (v)

- In (- y2(v) ) - In CX~)} CX

ap ff[ v=IZ1-ZI+P(v)(Z2-Z2)

(L P13/l 3Y + P(v)Lp23/l3y)J

[

_ -

p(v)

'\'

1...."

(743)

z~ )J

(739)

+

and

+

D:D~~0(

) cx(v») ytV) 2 D/lD~ + D:D y (cx(v)IY(v)1 - (l + CX(V»)ytV») im 2 n ( 1 + cx(v) * )! y(v) !2 - cx(v)y(v) * 2 P(3)y(v)

cx;v)pi~

1 2 cx(V)! y(v)1 - (l

(740)

- In (,- yt,j) - In

+

(1

G: :~:))]},

(745)

where CX;v) = (pt3) - p(v»)/2iPi~·

(746)

We have so far considered the most general case of anisotropy (with only one plane of elastic symmetry) which yields a completely analytical result. A less general but important case is when there is symmetry about three orthogonal planes at each point. This is the common case of orthotropy. This symmetry is exhibited, for

B. K D. Gairola

324

Ch.4

example, by a rolled plate which has symmetry planes parallel to the plane of the plate and parallel and perpendicular to the direction of rolling, forming a mutually orthogonal set of symmetry planes. The integrity bases for the crystal classes exhibiting this symmetry are listed in table 1 under the rhombic system. Using the procedure of Eshelby et al. [116J and Foreman [117J, Willis obtains for this case P(2)

= - I1 e

-jfJ

(747)

,

8

cg\ dCi~22'

-.1 -

(1) _

2

cos

(1) _ P22 p(l) a3

[2

*(2) _

Pll -P ll p(l) _ 12 -

-1

p(l) _ 21 -

*(2)

P22

= p(2) = a3

Ci;>13 /Cg)23'

2 C(2) C(2) C(2) ] (2 ) C(2) C 1122 1212 1122 1111 2222 2 C(2) (C(2) C(2) ) 1/2 ' 1212 1111 2222

e iO

I

. -(2-)4n 11 sm 28 [ C 1212 _ p*(2) _ 12 -

_

-

-

_ p*(2) _ 21 -

I .

,

p(3) afJ

C(2) 1122

C 1111 e (2)

(2)

C 2 2 2 2 C 12 1 2

= p(3) = a3

0

+

1212

+ -(2-)C

Xl = R,

2222

X

X k = X K (Z

,

2

+ v3)(J" + 3fl)B . 2 32n fleA + 2fl)R

(fl

ZD

l/ 2,

¢

=

tan -1

and

Z2/Z1

Z2 = R sin ¢,

Z

= Z3'

(A.I)

(A.2)

2

=

¢,

X

3

=

Z.

(A.3)

In general, we define curvilinear coordinates X Z, by

,

The result for the isotropic case can be obtained by performing a limiting operation on the result for rhombic orthotropic symmetry. In this way one obtains (A + fl + v2 + v 3 ) B InR -------,;:------"---- + 2 8n (J" + 2fl) R

+

The index notation used for general curvilinear coordinates employs superscripts instead of subscripts, e.g.

C(2)

(748)

2

(Zi

Zl = R cos ¢,

121 e iOJ

-i(j

=

or inversely

4n/i Ci~22 Cg)12 sin 28

(2 )

4n/ 1 sm 28.[ 0

1. Vectors and tensors in terms of natural base vectors

R

121 e -iOJ ~' C1111

+

325

functions of base vectors of a coordinate system. In the following we shall therefore develop tensor analysis in terms of base vectors. Many authors prefer to deal directly with tensor components because this procedure is more efficient. However, we feel that the former way helps in picturing the results.

+

-

-

I~ =

Nonlinear elastic problems

In the three-dimensional Euclidean space we can define a system of curvilinear coordinates by specifying three functions of Cartesian coordinates which have unique inverses. For instance cylindrical coordinates are specified by the equations

where

It =

§A.l

l ' Z2'

which has the unique inverse (A.5) 2

Appendix Tensor analysis in curvilinear coordinates As is well known, tensors may be regarded as invariant objects which are independent of the choice of coordinate system. However, they may also be defined as multilinear

3

We assume that the three functions Zjf.Y", X , X ) have continuous partial derivatives with respect to X k . This implies that the Jacobian of the transformation (A.6) In rectangular Cartesian coordinates we express a vector as a linear function and a tensor as a multilinear function of the unit base vectors ik • For example the position vector R is (A.7)

Acknowledgements The author expresses his most sincere gratitude to Prof. E. Kroner, who read the entire manuscript critically and gave numerous valuable suggestions for its improvement. The author would also like to thank Dr. C. Teodosiu for several useful discussions and for reading part of the manuscript.

in terms of Cartesian coordinates (A.4)

Z3),

(749)

This solution agrees only in the term containing (In R)/R with the previous solutions, e.g. eq. (352). This is not surprising since we have ignored the solution of the homogeneous equation corresponding to eq. (706).

K

and

oR

dR

= OZk dZk =

i dZ k'

(A.8)

Similar definitions can be given in curvilinear coordinates by introducing the natural base vectors. Suppose the same point P is specified by the curvilinear coordinates K X . Then it is obvious that R can be considered as a function of Xl , X 2 , X 3 and therefore we have dR --

oR axK

K

dX ,

Ch.4

B. K. D. Gairola

326

and the square of the line element is dS 2

= dR·dR = GKL dX K dX L,

(A.IO)

§A.l

Nonlinear elastic problems

where cKLM and f,KLM are equal to 1 if KLM is an even permutation of 123 and - 1 if KLM is an odd permutation of 123 and zero otherwise. We have also used the formula

where oR oR GKL = oXK' oX L

oR oX K

f,KLM eNPQ --

(A.11)

are called the metric coefficients of the coordinate system. The geometrical meaning of the vectors oRjoX K is simple; these are the natural base vectors directed tangentially to the X K coordinate curves, and as such they vary with the position. We set

327

b KN b Kp

bLN b Lp

bM N b Mp

K

L

M

b

b

Q

b

Q

(A.2l)

Q

Moreover, we can readily check that GK

= !(GlfKLMGL

GL

X

X

GM

(A.22)

or

= GK

(A.l2)

G M = (GlfKLM GK.

(A.23)

It is obvious from the definition of eq. (A.15) that

and rewrite eqs. (A.9) and (A.11) as

dR = G; dX K ,

(A.l3) (A.l4)

The base vectors GK are, in general, neither unit vectors, nor do they have the same physical dimensions. This can be seen by considering the example of cylindrical coordinates, For these coordinates IG1 1 = IG3 1 = 1, and IG2 1 = R. It is also clear that the GK are not, in general, orthogona1. Hence we can introduce three noncoplanar vectors reciprocal to GK : (A.15)

G K . GL

= bK L ·

(A.24)

In view of eq. (A.14) the above relation implies GK

= GKLGL,

(A.25)

where G KL satisfies the relation GKLG LM

= b KM ·

(A.26)

If we multiply eq.,(A.25) scalarly by G M we find that GK.G M

= G KM.

(A.27)

We can see now that a vector u can be expressed in terms of components in two ways: where G 1 x G2 etc. denote the vector product and [G 1 G2G3 J

= G 1·G2 x G3

u

(A.16)

is the triple scalar product which is numerically equal to the volume of the parallelepiped spanned by the vectors G 1 , G 2 , G 3 . It is easily verified that [G 1G2G3 J2

=

G 1·G1

G 1·G2

G 1·G3

G 2·G 1

G 2·G2

G 2·G3

G3 · G1

G 3 · G2

G 3 • G3

= G, (A.17)

where G = det (GKL). We can write eq. (A.15) more

1 G K = __ f,KLMG x 2-JG L

(A.18) r''''''1'Y'>''''''r'1''H

in the form

= GKU K = G K UK'

It should be noted that in curvilinear coordinates the summation convention applies only if one of the repeated indices is covariant and the other is contravariant. The components UK and UK are obviously not equal as they would be in Cartesian coordinates for which the base vectors ik and i k are the same. We distinguish them by using the terms contravariant and covariant components, respectively, for UK and UK' In the same way a tensor of arbitrary rank can be expressed as a multilinear function of base vectors GK and G K • For example, a second order tensor Tis gIven by

T = T KL G K G L = T KL GK GL = T KL GK G L.

or

(A.29)

If we multiply eq. (A.28) scalarly by GJ and G J respectively and use eqs. (A.14), (A.24) and (A.27) we find that UJ

(A.19)

(A.28)

= GJK UK

and

UJ

= G JK UK'

The quantities GKL and G KL thus have the same property as b KL because are used as the coefficients of a linear transformation operating on the contravariant components of a vector, they yield as a result, of the ,. ,.... contravariant or covariant components of the

, 0""

= - sin cjJi l + cos ¢i2 ,

(A.72.2)

G(3)

=

G

= i3 ,

(A.72.3)

z

Q1 R = Q24> =

= -

Ql4>

cos ¢,

(A.73.1)

= sin ¢,

(A.73.2) (A.73.3)

Q3Z = 1,

(A.65)

(A.69)

= cos ¢i l + sin ¢i2 ,

Q2R

I

(aU(K) ax

The base vectors GK are

(K)

where

a

G(K)

R = R cos ¢il + R sin ¢i2 + Zi3 ·

(A.63)

= ~KL'

In view of equation (A.60) we can express the operator VR in the form

vR =

(U(K) G(K»

(J)

This procedure can be generalized to tensors of any rank. As an illustration let us consider the cylindrical coordinates defined by eqs. (A.1) and (A.2). In this case the position vector R of a point P is given by

The relationship between the base vectors Cartesian system is given by 1

a ax

(J)

(A.59)

P(K)

=

M(L)'

(J)

and therefore we can also write

G(K)

(A.67)

aQ(K)M

Hence the derivative of a vector u is given by

_ -

= _1_,

=

ax- Q (J)

G(KK)

G(K)

(A.66)

r(J)(K)(L) G(L) .

(J)

It is easily seen that G(KK)

333

In analogy to eq. (A.35) we define

G(KK)

where the repeated indices inside the parentheses are not to be summed. Since in orthogonal coordinate systems GK L = 0 for K =1= L we can put P(K)

Nonlinear elastic problems

§A.3

1

r 4>4>R = - r 4>R4> = R

(A.74)

334

B. K. D. Gairola

and all other Q's and

Ch.4

r 's are zero. Therefore the gradient of a vector u is given by

OUR (lOUR U¢» VRu = oR GRGR + R o¢ - If GRG¢> oU¢>

(IOU¢>

OUR

+ OZ GRGZ

eu,

UR)

+ oR G¢>GR + R o¢ - Ii: G¢>G¢> + oZ G¢>Gz oU z

R

1 oU z

oU z

= oR GzG + R o¢ G z G¢> + oZ Gz Gz

(A.75)

and the divergence of a tensor T of second rank is

VR'T

=

OTRR 1 OT R¢> GR [ oR + R o¢ OTR¢>

OT RZ

1

]

1 OT¢>¢>

OT¢>z

1

]

1 o"'[¢>z

OT zz

1

+ oZ + R(TRR - T¢>¢»

+ Gc/> [ oR + R o¢ + oZ + R (TR¢> + Tc/>R) + G [OT RZ z

oR

]

+ Ii. o¢ + oZ + R T RZ .

(A.76)

Addendum (1976) Some remarks on the use ofnonlinear theory ofelasticity in the dislocation core problem

In subsect. 3.2 it may have been noticed by the reader that the solutions for the elastic fields depend critically on the core radius as well as on the core boundary conditions. In the core region the continuum theory breaks down and the atomic positions must be calculated on the basis of a discrete atomistic model. The usual procedure in such calculations has been to divide the dislocated crystal into two regions. The inner region contains the core where atom positions and interactions must be considered explicitly. In the outer region, the remainder of the crystal, the continuum theory is assumed to hold. In earlier models of this kind (for references see [123J) the atoms on the boundary between the two regions were held fixed in their position given by the linear theory of elasticity and the elastic solutions in their turn were obtained by neglecting the boundary condition in the core. These simplifying assumptions not only lead to stress discontinuities across the boundary but also induce artificial constraints such as the zero mean value of the volume dilatation. If the atomic configuration of the dislocation core were calculated with boundary conditions derived from the linear theory of elasticity, it would consist of a narrow dilated region around the dislocation line which would be compensated by an artificially compressed shell separating the inner core region from the elastic continuum. Improved solutions can be obtained if flexible boundary conditions are used [118, 119]. In this case the boundary atoms are allowed to move during relaxation and they are subjected not only to the forces from the atoms of the core but also to the forces due to the presence of atoms outside the core. These conditions, therefore,

Nonlinear elastic problems

335

allow an elastic deformation of the boundary and thus effectively increase the size of the core. Further improvement could be achieved if, as suggested by Seeger [106, 120J, the nonlinear elasticity theory were used in the outer region of the dislocation. A particular suggestion of Seeger (see [122J), is to take the stress vector on the boundary as a Fourier series of the polar angle with initially undetermined coefficients. As higher harmonics correspond to terms that vanish rapidly with the increasing distance from the dislocation core it is, in general, sufficient to consider only the first two or three harmonics. The solution of the nonlinear elasticity can be found by the iterative procedure described in subsect. 3.2.1. Moreover, the total energy (atomistically as well as elastically calculated energy) of the dislocation is minimized as a function of the undetermined coefficients occurring in the boundary conditions and of the displacement of the atoms located in the core. Recently Granzer et al. [121J have attempted to incorporate these ideas in calculations of the core configuration of edge dislocations in NaCI-type crystals. It can be seen that this method requires the knowledge of the elastic solution for the distortion produced by the dislocation under arbitrary core-boundary conditions. The linear anisotropic solution for the straight edge dislocation with such boundary conditions has been obtained by Teodosiu and Nicolae [122J using a complexvariable technique. The corresponding nonlinear anisotropic solution for the edge dislocation has been calculated by Seeger et al. [123J using the Signorini interaction scheme. They express the hope that this method will give a correct description of typical nonlinear effects and will provide an analytic solution applicable as close as two or three atomic distances from the centre of the dislocation core. On the other hand, Bullough and Sinclair [124J have numerically tested the accuracy of the solution obtained by Willis's perturbation scheme (as described in subsect. 3.6.3) for particular cases and they draw the conclusion that the usefulness of solutions obtained by using nonlinear elasticity is limited in the context of atomistic computer simulation. In the region near the dislocation where linear elasticity is no longer applicable the effects of discreteness and finite range are comparable to those of elastic nonlinearity. Thus at present this question remains open. Further developments in the nonlinear continuum theory ofdislocations

The concept of dislocations and disclinations has proved useful not only for studying crystalline solids but also for magnetic flux-line lattices in superconductors, high polymers and liquid crystals. Disclinations are characterized by Frank's rotation vector which is quite analogous to the Burgers vector of dislocations. Actual examples of dislocations and disclinations in flux-line lattices can be seen in figs. 9 and 10. The fact that the dislocation density (referred to the unit cell) in flux-line lattices is hundreds of times larger than the dislocation density commonly observed in metals leads to the conjecture that it plays an important role in determining the magnetic properties of superconductors. Polymers are built by long molecular chains. In a model discussed by Blasenbrey and Pechhold [127, 128J the molecular chains of the polymer melt join together to

jj6

B. K. D. Gairola

Ch.4

form bundles which are arranged in a meander-like structure (fig. 11). The meander model may be considered as a series of disclinations which are arranged in a sequence

Fig. 9.

Dislocations in the flux-line lattice in a superconductor, coupled to the flux-density gradients. (Photo: Essmann and Trauble [125].)

Fig. 10.

Wedge disclination in the flux-line lattice in a Type II superconductor. (Photo: Essmann and Trauble, from Anthony et al. [126].)

Nonlinear elastic problems

337

oflayers, the sign ofthe disclinations changing from one layer to the other. It should be mentioned, however, that the meander model is not generally accepted. Both the flux-line lattice and the high polymer structures can be looked upon as special cases of ordered line structures or rather line bundles. The term ordered line bundle is applied to any continuously varying bundle-type -arrangement of discrete lines which have no physical structure and which show a crystalline order in the plane perpendicular to the lines. The continuity is broken only at some points, lines and surfaces, and these singularities are called structural defects of the bundle. The lines are not always material as in polymers. For instance the magnetic fluxlines are non-material but physically real because the magnetic flux is essentially concentrated along the flux-lines which form, generally, two-dimensional hexagonal triangular lattices. It is, therefore, clear that neither can one introduce a material coordinate system nor can one define a lattice basis uniquely, in contrast to the point lattices where the lattice vectors are uniquely defined by the physical structure. Thus it is impossible to define the deformation commonly used in the deformation theory of point lattices. It follows that concepts of various non-Euclidean geometries cannot be used directly to describe the structural defects of the system. A nonlinear continuum theory of ordered line bundles has been developed recently [129-l31J which largely overcomes these difficulties by generalizing the concepts of nonEuclidean geometries. Since the lines of the bundle possess no inherent structure we may introduce a field of triads defining the bundle direction and joining neighbouring lines in an infinite number of ways. All the triads introduced in various ways are physically equivalent. Though each field of triads includes the essential physical

==t Fig. 11.

Meander model (Pechhold and Blasenbrey [128J).

s.

.J.JO

Nonlinear elastic problems

Ch.4

K. D. Gairota

information about the bundle configuration, it remains non-unique. Hence for a given configuration one can define an equivalence class of triads in the usual mathematical sense. Each bundle configuration thus uniquely corresponds to an equivalence class. In this way one arrives at the concept of "degenerate non-Euclidean geometry" which is capable of describing the physical structure ofordered line bundles. The main differences from the previous non-Euclidean theories consist in the use of equivalenceclass of displacement fields, nonregular metric (Det (gi) = 0, gijduj ~ 0 for u = 0), a degenerate connection, and correspondingly the degenerate torsion which is identified with the dislocation density.

Table 1 (continued) Crystal system

Hexagonal

Generating transformations

Integrity basis

m3m

R 3T1' M 2 , N

E nE[2 + E33E~1 + E 33E't3 + E llE;2 + E llE;l + E 22E't3' EllE;lE;2 + E22E;2E't3 + E 33E;3 E;1' E't3E22E33 + E~lE33Ell + E;2 Ell E22

3

S1

3

NS 1

E 33, Ell + E 22 , E llE22 - E;2' E ll[(Ell + 3E 22)2 - l2E;2J, E~l + E't3' E31(E~1 - 3E't3)' (Ell - E 22)E 31 - 2E12E23, (Ell - E 22)E 23 + 2E 12E 31' 3E12(E ll - E 22)2 - 4E[2' E 23(E;'3 - 3E~1)' EnE~1 + E llE;'3 - 2E23E31E12' E 31[(Ell + E 22)2 + 4(E;2 - E;'2)J - 8EllE12E23, E 23[(Ell + E 22)2 4(E;2 E't2)J + 8E llE 12E31, (Ell - E22)E23E31 + E 12(E;'3 - E;l)

32

s., D 1

3m

S1' R 1

32/m

R 1 , NS 1

6

R 3S 1

6

D 3S 1

6/m

NSl' R 3

6m2

R 3S1' R 1

622

D 3S 1,D 1

Type Crystal class

8

Table 1 Irreducible integrity bases for the strain tensor f for various crystal systems. (The third column gives the Hermann-Mauguin symbols for each class.) Generating transformations

Integrity basis

Crystal system

Type Crystal class

Triclinic

1

-

1

N

Monoclinic

2

2 m 2/m

D1 R1 N,R 1

Ell' E 22, E 33, E 23, E;3' E;2' E 13E 12

222 2mm mmm

n., D 2

Ell' E 22, E 33, E;3' E;3' E;2' E 12E23E 13

D 1, R 2 N, R 1 , R 2

4

D 1T 3

4 4/m

R 1T3 N, R 1T3

422 42m 4mm 4/mmm

R 1T3,D 1 D 1T3,D 1 R 1T 3 , R 1 N, R 1 , R 1T3

Ell + E 22, E 33, E;3 + E;3' E;2' E 11E22 E12E23E13' E llE;'3 + E 22E;3' E;3 E;'3

23

D 1,M 1

E 22 + E 33, E 22E 33 + E 33Ell + Ell E 22, EllE22E33' E;'3 + E;l + E;2' E;l E;2 + E;2 E't3 + E;'3E;1' E 23E31E12' E 22E;2 + E 33E;3 + E llE;l' E;l E33 + E;2 Ell + E;3 E22' E 33E;'2 + E llE;3 + E 22E;l' E;2 E:l + E;'3E{2 + E;1 Ei3' E llE;l E;2 + E 22E;2 E;'3 + E 33E't3 Eil' E;3 E22 E33 + Eil E33Ell + E;2 Ell E22' E llE22E;1 + E 22E 33E;2 + E 33E llE;3' E;'3Eil E22 + Eil E;2 E33 + E;2 E't3Ell

Orthorhombic 3

Tetragonal

4

5

Cubic

6

1

1

Ell' E 22, E 33, E 12, E 13 , E 23

Ell + E 22, E 33, E;3 + E;'3' E;211&E llE22, E 12(E 11 - E 22) E 13E23(E ll - E 22), E12E23E13' E 12(E;3 - E;'3) E llE;'3 + E 22E;3' E 13E23(E;3 - E;'3)' E;3 E;3

9

10

+

Ell

+

m3

7

N, D1' M 1

43m

D1' M 1 , T 1

432

R 3T 1 , M 2

Ell

+

E 22

+

E 33, E 22E 33 + E llE22, EllE22E33' E;3 + Eil + E;2' E;l E;2 + E;3 E;1' E23E31E12'

11

+

E 33E ll

6mm

D 3S1' R 1

+

E;2 E;3

6/mmm

D 3S 1 , R 1 , N

E 33, Ell + E 22, E llE22 - Ei2' E ll[(Ell + 3E22)2 - l2E;2J, Etl + E't3' E 23(E't3 - 3E~1)' (Ell - E 22)E 23 + 2E 12E31' E llE;l + E 22E;3 + 2E23E31E12' E 23[(Ell + E 22)2 4(E;'2 - E;2)J + 8E llE12E31 E 33, Ell + E 22, E llE22 - Ei2' E ll[(Ell + 3E22)2 - 12E;2J, E~1 + E't3' E~l(E~l - 3E't3?' E llE;'3 + E22E~1 - 2E23E31E12' E 12(Etl - E;'3) + (E 22 - Ell)E31E23' 3E 12(E 11 - E 22)2 - 4E{2' E 31E23[3(E;1 - E;'3)2 - 4EtlE;'3J, E ll(E:1 + 3Ei3) + 2E22E~1(E;1 + 3E;'3) - 8E12E23E;1' E;l[(E ll + E 22)2 - 4(E;2 - E;2)J - 2E ll[(Ell + 3E22)(E~1 + E't3) - 4E23E31E12J, E23E31[(Ell + E n)2 4(E;2 - E;2)J + 4E llE12(E;'3 - E;l)' E12[(E~1 + E;'3)2 + 4E'tiE~1 - E't3)J - 4EflE23(Ell - E 22) E 33, Ell + E 22, E llE22 - Ei2' E ll[(Ell + 3E22? - 12E;2J, Etl + E;3' Et1(Etl - 3E;'3)2, E llE;'3 + E nE;1 - 2E23E31E12' E ll(E:1 + 3Ei3) + 2E22E~1(E;1 + 3E;3) - 8E 12E23E;1' Etl[(E ll + E 22)2 -4(E;2 - E;2)] - 2E ll[(Ell + 3E22) x (E;l + E't3) - 4E23E31E12J

339

34U

B. K. D. 'Gairola

Ch.4

References [1] C. O'Neill, Mem. Mane. Lit. Phil. Soc. 1 (1862) 243. [2] H. O'Neill, J. Iron Steel Institute 109 (1924) 93. [3] L. M. C1arebrough, M. E. Hargreaves and G. W. West, Proc. Roy. Soc. A 232 (1955) 252; Phil. Mag. 1 (1956) 528; Acta Metall. 5 (1957) 738. [4] R. E. Peierls, Proc. Phys. Soc. Lond. 52 (1940) 34. [5] F. R. N. Nabarro, ibid. 59 (1957) 256; Theory of Crystal Dislocations (Clarendon Press, Oxford, 1967). [6] A. Seeger, in: Encyclopedia of Physics, S. Flugge (ed.) (Springer, Berlin, 1955) vol. VII/I, p. 383. [7] A. Seeger and P. Schiller, in: Physical Acoustics, W. P. Mason (ed.) (Academic Press, New YorkLondon, 1966) vol. III A, p. 361. [8] H. Bross, A. Seeger and P. Gruner, Ann. Phys. 11 (1963) 230. [9] H. Bross, A. Seeger and R. Haberkorn, Phys. Stat. Sol. 3 (1963) 1126. [10] A. Seeger andP. Brand, in: Small-Angle X-Ray Scattering, H. Brumberger (ed.) (Gordon and Breach, New York-London-Paris, 1967) p. 383. [11] F. D. Murnaghan, Finite Deformation of an Elastic Solid (Wiley, New York, 1953). [12] V. V. Novozilov, Foundations of the Nonlinear Theory of Elasticity, English translation (Gray1ock Press, Rochester N.Y., 1953). [13] A. E. Green and W. Zerna, Theoretical Elasticity (Clarendon Press, Oxford, 1954). [14] A. E. Green and J. E. Adkins, Large Elastic Deformations and Nonlinear Continuum Mechanics (Clarendon Press, Oxford, 1960). [15] C. Truesdell and R. A. Toupin, in: Encyclopedia of Physics, S. Fliigge (ed.) (Springer, Berlin, 1960) vol. III/I, p. 226. [16] A. C. Eringen, Nonlinear Theory of Continuous Media (McGraw-Hill, New York, 1962). [17] C. Truesdell and W. Noll, in: Encyclopedia of Physics, S. Flugge (ed.) (Springer, Berlin, 1965) vol. III/3. [18] A. Seeger and E. Mann, Z. Naturforschg. 14a (1959) 154. [19] Z. Wesolowski and A. Seeger, in: Mechanics of Generalized Continua, E. Kroner (ed.) (Springer, Berlin-Heidelberg-New York, 1968) p. 295. [20] A. Seeger and Z. Wesolowski, in: Physics of Strength and Plasticity, A. S. Argon (ed.) (M.LT. Press, Cambridge (Mass.)-London, 1969) p. 15. [21] K. Kondo, Proc. 2nd Japan Nat. Congr. Appl. Mech., 41 (1952); RAAG Memoirs, 1-4 (1955-1967). [22] B. A. Bilby, R. Bullough and E. Smith, Proc. Roy. Soc. A231 (1956) 481. [23] E. Kroner and A. Seeger, Arch. Rational Mech. Analysis 3 (1959) 97. [24] E. Kroner, Arch. Rational Mech. Analysis 4 (1960) 273. [25] H. Pfleiderer, A. Seeger and E. Kroner, Z. Naturforschg. 15a (1960) 758. [26] B. A. Bilby, L. R. T. Gardner and A. N. Stroh, Extrait des Actes, 9 Congr. Int. Mec. Appl. 8 (1957) 35. [27] R. Stojanovitch, Phys. Stat. Sol. 2 (1962) 566. [28] R. Bullough and J. A. Simmons, in: Physics of Strength and Plasticity, A. S. Argon (ed.), (M.I.T. Press, Cambridge (Mass.)-London, 1969) p. 47. [29] J. R. Willis, Int. J. Engng. Sci. 5 (1967) 171. [30] C. Teodosiu and A. Seeger, in: Fundamental Aspects of Dislocation Theory, J. A. Simmons, R. de Wit and R. Bullough (eds.), Nat. Bur. Stand. (U.S.), Spec. Publ. 317, II, 1970. [31] H. Hencky, Z. Techn. Phys. 9 (1928) 214. 457: Z. Phys. 55 (1929) 145; Ann. der Phys. (5), 2 (1929) 617. [32] Z. Karni and M. Reiner, Bull. Res. Counc. of Israel 8c (1960) 89; in: Second-Order Effects in Elasticity, Plasticity and Fluid Dynamics, M. Reiner and D. Abir (eds.) (Macmillan, New York, 1964) p.217. [33] B. R. Seth, ibid. p. 162. [34] D. B. MacVean, ZAMP 19 (1968) 157. [35] F. D. Murnaghan, Amer. J. Math. 59 (1937) 235. [36] J. L. Ericksen, in: Encyclopedia of Physics, S. Flugge (ed.) (Springer, Berlin, 1960) vol. III/I, p. 794. [371 M. Lagally, Vorlesungen iiber Vektor-Rechnung (Akad. Verlagsges., Leipzig, 1945). [381 E. A1mansi. III. Rend. Lincei (5A) 20 (1911) 287.

Nonttnear etastic problems

j41

[39] G. Hamel, Elementare Mechanik (B. G. Teubner, Leipzig & Berlin, 1912). [40] E. J. Nanson, Mess. Math. 7 (1877-1878) 182-185. [41J L. Euler, Misc. Taur. 2 (1760-1761) 1-10; Opera omnia (2),100762) 255-263; Oevres de Lagrange 14(1762) 178-188. [42] A. L. Cauchy, Ex. de Math. 2 (1827) 108-111; Oevres (2),7 (1827) 141-145. [43] V. D. Bondar, PMM 33 (1969) 1094. [44] E. Kroner, Z. Phys. 142 (1955) 463. [45] G. Kirchhoff, J. reine angew. Math. 56 (1859) 285: Vorlesungen tiber Mathematische Physik: Mechanik (B. G. Teubner, Leipzig, 2nd ed. 1877, 3rd ed. 1883). [46] A.-J.-C. B. de St. Venant, Resume des Lecons sur l'app1ication de la Mecanique (par C.-L.-M.-H. Navier), Appendix 3 (Didot, Paris, 1864). [47] J. Boussinesq, J. Math. Pures Appl. (2) 16 (1871) 125. [48J F. Odqvist, C.R. Acad. Sci. (Paris) 205 (1937) 1128. [49] V. Z. Vlasov, PMM 8 (1944) 301. [50] G. Piola, Opusc. mat. fisc. di diversi autori, Milano: Giusti (1833) 201. [51] G. Kirchhoff, Sitzgber. Akad. Wiss. Wien 9 (1852) 762. [52] R. S. Rivlin, in: Second-Order Effects in Elasticity, Plasticity, and Fluid Dynamics, M. Reiner and D. Abir (eds.), (Macmillan, New York, 1964) p. 198. [53J J. Boussinesq, Compt. Rend. Acad. Sci. Paris 71 (1870) 400; Mem. Divers Savants 20 (1872) 509. [54] W. Thomson (Lord Kelvin), Phil. Trans. Roy. Soc. London, (A)153 (1863) 583. [55] E. and F. Cosserat, Ann. Toulouse 10 (1896) 1. [56] C. Neumann, R. Reine Angew. Math. 57 (1860) 281. [57] R. N. Thurston and K. Brugger, Phys. Rev. 133 (1964) A1604. [58] H. Wey1, The Classical Groups (Princeton University Press, 1939). [59] G. F. SmiTh and k. s: Riv1i~, Trans. Am. Math. Soc. 88 (1958) 175. [60] G. F. Smith, Arch. Rational Mech. Anal. 10 (1962) 108. [61] S. Bhagvantam and D. Suryanarayana, Acta Cryst. 2 (1949) 21. [62] H. A. Jahn, ibid. 2 (1949) 30. [63] F. G. Fumi, Phys. Rev. 83 (1951) 1274; 86 (1952) 561. [64] R. F. S. Hearmon, Acta Cryst. 6 (1953) 331. [65] T. S. G. Krishnamurty, ibid. 16 (1963) 839. [66] F. C. Frank, Phil. Mag. 42 (1951) 809. [67] W. T. Read, Jr., Dislocations in Crystals (McGraw-Hill, New York, 1953). [68] Z. Knesl and F. Seme1a, ZAMM 52 (1972) 381. [69] H. Kauderer, Nichtlineare Mechanik (Springer, Berlin, 1958). [70] W. Thomson (Lord Kelvin) and P. G. Tait, Treatise on Natural Philosophy, Part I (Cambridge, 1867). [71] J. H. Poynting, Phil. Mag. 9 (1905) 393 = Papers 2,335; Proc. Roy. Soc. Lond. A82 (1909) 546 = Papers 2,358; India-Rubber J., Oct., 4 (1913) p. 6 = Papers 2, 424. [72] R. Gramme1, Actes du Colloque International des Vibrations Non-Lineaires (1951), Publ. Scient. et Techn. du Min. de I'Air, no 281 (Paris, 1953) p. 45. [73] F. Granzer, Acta Metall. 18 (1970) 159. [74] H. Suzuki, Nachr. Akad. Wiss. Gottingen, II Math. Phys. K1asse, Nr. 6 (1971). [75] A. Signorini, Proc. 3rd Int. Congr. Appl. Mech. 2 (1930) 80; Atti 24th Riun. Soc. Ital. Prozi. Sci. 3 (1936) 6; Atti Cony. Mat. Roma 1942, p. 153; Ann. Mat. pura appl. 30 (1949) 10. [76] M. Misicu, Stud. Ceret. Mec. Met. 4 (1953) 31. [77] A. E. Green and E. B. Spratt, Proc. Roy. Soc. A224 (1954) 347. [78] R. S.Riv1in and C. Topakog1u, J. Rational Mech. Anal. 3 (1954) 581. [79] D. A. Da Silva, Mem. Ac. Sc. Lisboa (2a), 3 (1851) 61. [80J N.1. Muskhelishvi1i, Some Basic Problems of the Theory of Elasticity (Noordhoff, Groningen, 1953). [81] G. Lame, Lecon sur 1a Theorie Mathematique de l'Elasticite, 2nd ed. (Bachelier, Paris, 1866). [82J C. Teodosiu, to be published. [83J R. S. Riv1in,Phi1. Trans. Roy. Soc., A240 (1948) 459, 491, 509; A241 (1948) 379; A242 (1949) 173; Proc. Camb. Phil. Soc. Math. Phys. Sci. 45 (1949) 485; Proc. Roy. Soc. Al95 (1949) 463. [841 J. E. Ericksen, ZAMP 5 (1954) 466: J. Math. Phys. 34 (1955) 126.

LJ • ..1\.... .1../.

\JUlfUlU

Ln. 4

[85] W. Klingbeil and R. T. Shield, ZAMP 17 (1966) 489. [86] M.Singh and A. C. Pipkin, ZAMP 16 (1965) 706. [87] R. L. Fosdick, in: Modern Developments in Mechanics of Continua, S. Eskinazi (ed.) (Academic Press, London-New York, 1966). [88] A. E. Green, R. S. Riv1in and R. T. Shield, Proc. Roy. Soc., A211 (1952) 128. [89] G. Albenga, Atti Accad. Sci., Torino, cl. sci. fl. mat. natur. 54 (1918/19) 864. [90] G. Colonnetti, Atti Accad. naz. Lincei Rc. 27/2 (1918) 155. [91] E. Kroner, Kontinuumstheorie der Versetzungen und Eigenspannungen (Springer, Berlin, 1958). [92] C. Zener, Trans. A1.M.E. 147 (1942) 361. [93] R. W. Keyes, Acta Metall. 6 (1958) 611. [94] A. Seeger and P. Haasen, Phil. Mag. 3 (1958) 470. [95] A. Seeger, Nuovo Cim. 7 (1958) 632. [96] C.-C. Wang, ZAMM 46 (1966) 141. [97] R. A. Toupin and R. S. Riv1in, J. Math. Phys. 1 (1960) 8. [98] E. Kroner and G. Rieder, Z. Phys. 145 (1956) 424. [99] J. F. Nye, Acta Metall. 1 (1953) 153. [100] W. Noll, Arch. Rational Mech. Anal. 27 (1967) 1. [101] S. Moriguti, Oyo Sugaki Rikigaku (Appl. Math. Mech.) 1 (1947) 29,87. [102] C. Eckart, Phys. Rev. 73 (1948) 373. [103] J. A. Schouten, Ricci-Calculus (Springer, Berlin, 1954). [104] E. Cartan, Lecons sur la geometric des espaces de Riemann (Gauthier-Villars, Paris, 1928). [105] E. Kroner (ed.), Mechanics of Generalized Continua (Springer, Berlin-Heidelberg-New York, 1968). [106] J. A. Simmons, R. de Wit and R. Bullough (eds.), Fundamental Aspects of Dislocation Theory, Nat. Bur. Stand. (U.S.), Spec. Publ. 317, II (1970). [107] R. F. Gwyther, Mem. Manchester Lit. Phil. Soc. 56, No. 10 (1912). [108] B. Finzi, Rend. Lincei (6) 19 (1934) 578, 620. [109] E. Beltrami, ibid. (5) 1 (1892) 141. [110] J. H. Michell, Proc. Lond. Math. Soc. 31 (1899) 100, 130. [Ill] E. Kroner, Z. Phys. 139 (1954) 175; 143 (1955) 374. [112] K. Marguerre, ZAMM 35 (1955) 242. [113] V. L. Indenbom and S. S. Orlov, PMM 32 (1968) 414. [114] J. R. Willis, Phil. Mag. 21 (1970) 931. [115] T. Mura and N. Kinoshita, Phys. Stat. Sol. (b) 47 (1971) 607. [116] J. D. Eshelby, W. T. Read and W. Shockley, Acta Metall. 1 (1953) 251. [117] A. J. E. Foreman, ibid. 4 (1955) 322. [118] J. F. Sinclair, J. Appl. Phys. 42 (1971) 5321. [119] P. C. Gehlen, J. P. Hirth, R. G. Hoagland and M. F. Kanninen, J. Appl. Phys. 43 (1972) 3921. [120] A Seeger, in: Interatomic Potentials and Simulation of Lattice Defects. P: 566, 764-65 (Battelle Institute Material Science Colloquia, June 1971). [121] F. Granzer, V. Belzner, M. Bucher, B. Petrasch and C. Teodosiu, J. Physique 34 (Colloque C9, suppl. 11/12) c 9-359 (1973). [122] C. Teodosiu and V. Nico1ae, Rev. Roumaine Sci. tech., ser. mec, appl. 17 (1972) 919. [123] A. Seeger, C. Teodosiu and P. Petrasch, Phys. stat. sol. (b), 67 (1975) 207. [124] R. Bullough and J. F. Sinclair, Atomic Energy Research Establishment, Progress Report/Tf' 29 (1974). [125] U. Essmann and H. Trauble, Phys. stat. sol. 32 (1969) 337. [126] K. H. Anthony, D. Essmann, A. Seeger and H. Trauble, Disclinations and the Cosserat Continuum with Incompatible Rotations in ref. [105]. [127] S. Blasenbrey and W. Pechho1d, Ber. Bunsenges. Physik. Chem. 74 (1970) 784. [128] W. Pechhold and S. Blasenbrey, Kolloid-Z. und Z.-Polymere, 241 (1970) 955. [129] K. H. Anthony, Habilitationsschrift, Universitat Stuttgart (1974). [130] E. Kroner and K. H. Anthony, Dislocations and Disclinations in Material Structures: the basic Topological Concepts, in Annual Reviews of Material Science, Vol. 5 (1975). [131] K. H. Anthony, Arch. Mech. 28, No.4 (1976).

Author Index Aderogba, K., 192,219 Adkins, J. E., 225, 260, 276, 340 Aerts, E., 32 Albenga, G., 287, 342 A1mansi, E., 240, 341 Al'shitz, V. A., 110, 141 Amari, S., 99, 139 Ame1inckx, S., 32, 211, 212, 220, 221 Anthony, K. H., 336, 337,342 Ardell, A. J., 197,220 Ashby, M. F., 209, 211, 220 Atkinson, c., 218, 221 Bacon, D. J.,.154, 162,165, 199,200,208,220 Bardeen, J., 80, 139 Barenblatt, G. 1., 85, 139 Barnett, D. M., 150, 164,216,219,221 Barrand, P., 209,211,220 Bastecka, J., 201, 220 Beltrami, E., 308,342 Belzner, V., 335,342 Bhagvantam, S., 263, 341 Bilby, B. A., 160,165, 196,220,226,263,302,340 Blasenbrey, S., 335,337,342 Blokh, V. 1., 209, 220 Bocek, M., 134, 140 Boiko, V. S., 83, 85, 136, 137, 139, 140, 141 Bondar, V. D., 246,341 Bordoni, P. G., 32 Bouligand, Y., 16,31 Boussinesq, J., 248, 257, 341 Bramble, J. H., 209,220 Brand, P., 225, 340 Bross, H., 225, 340 Brown, L. M., 145, 153, 154,164,165 Brown, W. F., 50, 138, 188,219 Brugger, K., 259, 341 Bucher, M., 335, 342 Bullough, R., 132, 140, 154, 162, 165, 209, 211, 220, 226, 263, 302, 335, 340, 342 Burgers, J. M., 31, 47, 48, 50, 138 Burkanov, A. N., 137, 141 Burnside, W. S., 195,219

Cartan, E., 300, 342 Cauchy, A. L., 242, 341 Cheng, D. H., 199,220 Chernizer, G. M., 110, 139 Chishko, K. A., 137,141 Chou, Y. T., 159, 165 C1adis, P., 16,31 C1arebrough, L. M., 225, 288, 340 Clements, D. L., 218, 221 Collins, W. D., 209, 220 Colonnetti, G., 287, 342 Comninou, M., 218, 221 Corbino, O. M., 186,219 Cosserat, E., 258, 341 Cosserat, F., 258, 341 Cottrell, A. H., 32, 54, 138, 160,165 Coulomb, P., 175,209,219,220 Cruse, T. A., 170, 219 Da Silva, D. A., 270, 341 Dedukh, L. M., 132, 133, 140 De Gennes, P. G., 15, 16,31,32 Deh1inger, D., 20, 32 De1avignette, P., 32, 211, 212, 220, 221 De St. Venant, A.-J.-C. B., 248, 341 De Wit, R., 92, 126, 139, 302, 335, 342 Dietze, H.-D., 181,219 Doring, W., 31 Dubnova, G. N., 68, 79, 138, 139,152,154,164 Dubrovskii, 1. M., 136, 141 Dundurs,J., 175, 190, 192, 193, 196, 198,201,208, 209,218,219,220,221 Eckart, c., 297, 342 Ericksen, J. E., 275, 341 Ericksen, J. L., 233, 340 Eringen, A c., 225, 340 Eshelby, J. D., 52, 72, 85, 87, 103, 108, 110, 111, 113, 115, 116, 117, 118, 119, 138, 139, 140, 155, 165,173,176,177,183,196,201,204,211,215, 216,219,220,221,320,323,324,342 Essmann, D., 336, 342 Euler, L., 242, 341

Autnor tnaex

riui nor tnaex

Fel'dman, E. P., 52, 138, 195,219 Feldtkeller, E., 16,31 Finzi, B., 306, 342 Foreman, A. J. E., 52, 138, 155, 165, 324, 342 Forwood, C. T., 164, 165 Fosdick, R. L., 275, 342 France, L. K., 163, 165 Frank, F. c., 31, 32, 54,68, 72, 77, 108,138,139, 215,221,263,341 Frank, Ph., 187, 219 Franz, H., 94, 139 Frenkel, J., 21, 32 Frenkel, Ya. I., ll5, 140 Friedel, G., 32 Friedel, J., 32, 122, 140, 175, 219 Fumi, F. G., 263, 341 Garber, R. I., 136, 137, 141 Gardner, L. R. T., 226,340 Gavazza, S. D., 216, 219, 221 Gebbia, M., 169, 219 Gehlen, P. c., 334,342 Gemperlova, J., 195, 219 Gendelev, S. Sh., 132, 140 Gomer, R., 215,221 Grammel, R., 266, 341 Granato, A., ll6, 140 Granato, A. V., 137,141 Granzer, F., 266, 335,341,342 Green, A. E., 208, 220, 225, 260, 267, 276, 280, 340,341,342

Greenhill, A. G., 183,219 Groves, P. P., 199,200,208,220 Gruner, P., 225, 340 Gwyther, R. F., 306,342 Haasen, P., 288, 342 Haberkorn, R., 225,340 Hahn, G. T., 135, 140 Hamel, G., 240, 259, 341 Hargreaves, M. E., 225, 288, 340 Hartley, C. S., 163, 165 Hayns, M. R., 209, 211,220 Head, A K., 135,136,140,156,163,165, 175,193, 195,208,219,220 Hearmon, R. F. S., 263,341 Heidenreich, R. D., 22, 32 Hencky, H., 229,340 Herring, c., 80, 139 Hetenyi, M., 208, 220 Hirsch, P. B., 206, 220 Hirth, J. P., 163,165,215,218,221,334,342 Hoagland, R. G., 334,342 Hollander, E. E., 95, 99, 139

Hsieh, C. F., 209, 220 Humble, P., 164,165 Indenbom, V. L., 68, 79, 86, 87, 99, llO, 132, 138, 139,140, 141, 145, 152, 153, 154, 164, 320,342 Iosilevskii, Ya. A., 136,141 Jahn, H. A., 263, 341 Jossang, T., 163, 165 Kaganov, M. I., 138, 141 Kanninen, M. F., 135,140,334,342 Karni, Z., 229, 340 Kauderer, H., 266, 341 Keyes, R. W., 288,342 Kinoshita, N., 320, 342 Kirchhoff, G., 248, 251,341 Kiusalaas, J., 103, 139 Kivshik, V. F., 136, 137,141 Kleman, M., 15, 16, 18,31,32 Klingbeil, W., 275, 342 Knesl, Z., 266, 341 Kober, H., 181,219 Koehler, J. S., 48, 50, 64, 101, 116, 138, 140, 188, 219 Kondo, K., 226, 302, 340 Kontorova, T. A., 115, 140 Kontorowa, T., 21, 32 Kosevich,A. M.,52,83,85,86,94,95,97,99, 100, 101, 102, 103, 106, 109, Ill, ll9, 120, 130, 131, 135, 136, 138, 139, 140, 141, 195,219 Kosevich, V. M., 52, 138, 195,219 Kovalev, A. S., 136,141 Kravchenko, C. Ya., 138,141 Krishnamurty, T. S. G., 263, 341 Krivenko, L. F., 136, 137, 141 Krivulya, S. S., 137, 141 Kroner, E., 32,48,87,90,92,94,96, 138, 139,226, 227,247,248,287,294,295,300,302,308,309, 313,337,340,341,342 Kroupa, F., 66, 89,101,138,139,212,220

Kunin, I. A, 91, 139 Kurihara, T., 198,220 Lagally, M., 239, 340 Lame, G., 273, 341 Landau, L. D., 53, 72, 85, 108, 138 Lee, M.-S., 196, 198,220 Lehmann, 0., 32 Leibfried, G., 48, 72, 115, 138, 140, 181,219 Lekhnitskii, S. G., 155, 165 Li, J. C. M., 69, 138 Lidiard, A. B., 204, 220

Lifshitz, E. M., 53, 72, 85, 108, 138 Lifshitz, 1. M., 48, 136, 138, 141 Lin, T. H., 199,220 List, R. D., 192,219 Lothe, J., 145, 152, 153,164 Love, A. E. H., 50, 102, 117, 138, 170, 186, 187, 196, 198, 199,201,208,209,219 Lubov, B. Ya., 110, 139 Lucke, K., 116, 140 Lur'e, A. I., 199,220 Lyubov, B. Ya., 135,140 MacVean, D. B., 229, 340 Malen, K., 104, 139 Mann, E., 226, 265,270, 288, 340 Mann, E. H., 212, 221 Maradudin, A. A., 136,141 Marguerre, K., 308, 342 Margvelashvili.T, G., 130, 140 Maruyama, T., 199,220 Mason, W., 110,139 Melan, E., 199, 220 Meyer, R. B., 32 Michel, L., 32 Michell, J. H., 308,342 Milne-Thomson, L. M., 183,219 Mindlin, R. D., 199,220 Misicu, M., 267, 341 Mitchell, L. H., 208, 220 Morgan, R., 163,165 Moriguti, S., 297, 309,342 Morton, A. J., 163,165 Mott, N. F., 32 Mura, T.,95, 99,103,139,148,164,192,219,320, 342 Murnaghan, F. D., 225, 231, 240, 340 Musher, J. 1., 197,220 Muskhe1ishvili, N. 1., 72, 73, 138, 273, 341 Nabarro, F. R. N., 18,32, 37,49, 72, 77, 85, 87, 99, 108, 125, 138, 139, 140, 161, 165, 182,209, 211,214,215,216,219,220,225,340

Nadgornyi, E. M., 135,140 Nanson, E. J., 242,341 Natsik, V. D., 100, 102, 103, 106, 107, 111, 119, 120, 137, 138, 139, 140, 141 Neel, L., 32 Nehari, Z., 178, 180,219 Neumann, c., 259, 341 Newey, C. W. A., 204,220 Newman, R. C., 209, 220 Nicholson, R. B., 197,220 Nicolae, V., 335, 342 Nikitenko, V. 1., 132, 133, 140

j"f:J

Noll, W., 225, 254, 291, 297, 340,342 Novozilov, V. V., 225, 340 Nye, J. F., 296, 342 Odqvist, F., 248, 341 Ogawa, K., 84, 139 O'Neill, c., 225, 288, 340 O'Neill, H., 225, 288, 340 Orlov, A. N., 79, 86, 87,99,139 Orlov, S. S., 145, 152, 153,164,320,342 Otruba, J., 163, 165 Panton, A. W., 195,219 Pastur, L. A., 52, 138, 195,219 Pati, S. R., 209, 211, 220 Peach, M., 48, 64, 101, 138 Pechhold, W., 335, 337, 342 Peierls, R., 20, 32 Peierls, R. E., 75, 138, 225, 340 Petrasch, B., 334, 335, 342 Pfleiderer, H., 226, 227, 302, 340 Piola, G., 251, 341 Pipkin, A. c., 275, 342 Poynting, J. H., 266, 341 Pratt, P. L., 204, 220 Preininger, D., 134,140 Ralph, B., 163, 165 Rault, J., 16,32 Read, W. T., 52, 54, 77, 138, 155, 165, 320, 323, 324,342 Read, Jr., W. T., 263, 341 Reid, C. N., 163, 165 Reiner, M., 229, 340 Rice, J. R., 218, 221 Riedel, H., 218, 221 Rieder, G., 96, 139, 294, 342 Rivlin, R. S., 254, 260, 267,275, 276, 288, 294, 388, 341,342 Rogula, D., 104, Ill, 112,139 Roitburd, A. L., 123, 128, 140 Rongved, L., 208, 220 Rosenfield, A R., 135, 136, 140 Rozentsveig, L. N., 48, 138 Salamon, N. J., 201, 208,220 Saralidze, Z. K., 130, 131, 140 Saxl, 1., 163, 165 Schiller, P., 225, 340 Schock, G., 52, 138 Schoeck, c., 110, 113,139 Schoeck, G., 32 Schouten, J. A., 297, 342 Schwenker, R. 0., 137,141

Seeger, A., 32, 52, 110, 113, 114, 138, 139, 188, 190,192,196,219,225,226,227,247,265,270, 275,288,302,334,335,336,340,342 Semela, F., 266, 341 Sendeckyj, G. P., 190, 192, 193,219 Sergeeva, G. G., 120,140 Seth, B. R., 229, 340 Shcherbak, N. G., 132,140 Shield, R. T., 275, 276,342 Shockley, W., 22,32, 52, 54, 138, 155, 165, 320, 323, 324, 342 Shtolberg, A. A., 79, 139 Signorini, A., 267, 341 Siems, R., 183,211,212,219,220,221 Simmons, J. A, 226, 302, 335, 340, 342 Sinclair, J. F., 334, 335,342 Singh, M., 275, 342 Slezov, V. V., 130, 131,140 Slutskin, A. A., 120, 140 Smith, D. A., 163,165 Smith, E., 226, 263, 302, 340 Smith, G. F., 260, 338, 341 Sokolnikoff, 1. S., 213, 214,219 Solomon, L., 199, 220 Solovev, V. A., 134, 135, 140, 218, 221 Southwell, R. V., 216,221 Spence, G. B., 182,219 Spratt, E. B., 267, 341 Steeds, J., 206, 220 Steeds, J. W., 145, 156, 158, 161,164,165 Steketee, J. A, 198,220 Stippes, M., 198, 220 Stojanovitch, R., 226, 340 Stroh, AN., 52, 72, 73, 138, 201, 204, 218, 220, 221,226,340 Suryanarayana, D., 263,341 Suzuki, H., 266, 341 Swanger, L. A., 150, 164 Tait, P. G., 266, 341 Teodosiu, c., 226, 275, 334, 335,340,341,342 Tewary, V. K., 136, 141 Thompson, R. M., 218, 221 Thomson, W. (Lord Kelvin), 258, 266, 341

Thurston, R. N., 259,341 Tomate, 0., 218, 221 Tomilovskii, G. E., 132, 140 Topakoglu, C., 267, 341 Toulouse, G., 18,32 Toupin, R. A, 225, 246, 276, 288, 294, 340, 342 Trauble, H., 336, 342 Truesdell, c., 225, 246,254,276, 291, 340 Tucker, M. 0., 195,220 Tung, T. K., 199,220 Tunstall, W. J., 206, 220 V. Laue, M., 180,219 V. Mises, R., 187,219 Van Bueren, H. G., 79, 139 Vershok, B. A., 123,140 Vitek, V., 218, 221 Vladimirskiy, K. V., 81, 82,139 Vlasov, V. Z., 248, 341 Volterra, E., 186,219 Volterra, V., 20, 32, 147,164, 186,219 Vreeland, T., 204, 220 Wagoner, R. H., 218, 221 Wang, cc., 288,342 Weeks, R., 198, 220 Weeks, R. W., 209, 211,220 Weertman, J., 64, 135,138,140 Weingarten, G., 32 Wesolowski, Z., 226, 265,275,340 West, G. W., 225, 288, 340 Westergaard, H. M., 199,220 Weyl, H., 260, 341 Willis, J. R., 145, 148, 149, 150, 151, 154, 162,164, 165,209,211,220,226,303,320,340,342 Willmore, T. J., 208, 220 Wood, W. W., 136,140 Yoffe, E. H., 202, 218, 219, 220 Zaitsev, S. 1., 135, 140 Zener, C., 110, 111, 139,288,342 Zerna, W., 225,276, 280, 340

Subject Index Airy stress function, 183 angular dislocations, 153, 202, 218 anisotropic medium, 8 approximate solutions for boundary problems, 211 arrays, movement of, 134 Bardeeri-Herring source, 80 bending ofrod, 212 bent dislocations, 163 Burgers vector, 12, 39,40 Burgers vector, local, 263 Burgers vector, true, 263, 295, 300 Cauchy stress, relation to Eulerian strain, 303 Cauchy stress tensor, 251,253 Cauchy's deformation tensor, 229, 233, 236 Cauchy's equations of motion, 253 Cayley-Hamilton equation, 239 climb, 6,58 climb, speed of, 121 climb of edge dislocation, 161 coaxial circles, theory of, 172 coherent twin boundary, 81 Colonetti's theorem, 211 compatibility equations, nonlinear theory, 243 condensation (of point defects), 121 conformal mapping, 179 constitutive equations, nonlinear theory, 254, 303 continuous distributions, 85 continuous distributions. movement of, 135 continuous distributions; nonlinear theory, 294 contour integrals in solutions for anisotropic media, 150 convected coordinates, 236, 280 core, atomic description of, 23 core, energy of, 53 core, radius of, 20, 53 core, splitting of, 22 core correction, 154 core cut-off, 20 core of dislocation, 7, 20, 38

core region, nonlinear theory in, 334 Cottrell atmosphere, 118 covalent solids, 28 covariant differentiation, 328 crack, dislocation near, 196,218 cracks, 85 cross slip, 58 cross slip, double, 79 cross slip of screw dislocation, 162 crystal symmetry and simplification of formulae, 156 crystals, 10 curvature tensor, 245 curvilinear coordinate system, 231, 325 curvilinear coordinate system, moving, 234 curvilinear coordinate system, orthogonal, 331 damping, 109, 138 deformation, change of area on, 241 deformation, change of volume on, 242 deformation, compatibility of, 245 deformation, nonlinear theory, 227 deformation gradient, tensor of, nonlinear theory, 228 deformation tensor, Cauchy's, 229, 233, 236 deformation tensor, Green's, 229, 233, 236 deformations, controllable, 275, 282 diffusion, 58, 121 dilatation field of screw dislocation, 159, 160, 164 dipoles, 67, 163 disclinations, 13, 335 dislocation density, local, nonlinear theory, 296 dislocation density, relationship to incompatibility, nonlinear theory, 297 dislocation density, true, 295 dislocation density tensor, 86 dislocation loop, self-energy of, 92 dislocation loops, 40, 41, 57 dislocation loops, force on, 62 dislocation loops, kinetic energy of, 107 dislocation mill, 80 dislocation moment tensor, 49

IJfA,UJLt.-t.

dislocation motion, 55,94, 134 dislocation polarization tensor, 89, 100 dislocation sources, 77 displacement field, 46, 147 displacement field of dislocation loop, 200, 208, 209 displacement field of screw dislocation, 157, 171, 173,201 displacement vector, 38,41, 146 displacement vector, nonlinear theory, 230, 233 distortion tensor, nonlinear theory, 295 easy glide (stage I), 24 edge dislocation in circular cylinder, 183 edge dislocation in circular disk, 207 edge dislocation in plate, 207 edge dislocation near hole, 190,218 edge dislocations, 38 edge dislocations, climb of, 121, 161 edge dislocations, field of, 49, 158 edge dislocations, formation of, 42 edge dislocations, glide of, 161 edge dislocations in plate, 196,211 edge dislocations near boundaries, 183, 195 effective mass, 107 Einstein tensor, 247 elastic centre of dilatation, 117 elastic constants, independent, nonlinear theory, 262 elastic distortion tensor, 85 elastic energy of grain boundary, 70 elastic energy of straight dislocation, 53 elastic field, interaction with dislocations, 62 elastic field, three-dimensional, 198, 218 elastic field, two-dimensional, 171 elastic field in presence of spherical boundary, 209,219 elastic field of moving dislocations, 98 elastic field of straight dislocation, 52 elastic field of wall, 69 elastic inhomogeneity, 169 elastic medium, 3 elastic solutions for continuous distributions of dislocations, nonlinear theory, method of Kroner et aI., 306 elastic solutions for continuous distributions of dislocations, nonlinear theory, method of Willis, 316 elastic solutions for dislocation, nonlinear theory, 265 electron microscopy, dislocation image widths, 164 electrostatic analogy, 180 energy factor, 153, 155

Subject index

l-ft.WCA

energy of edge dislocation, 186 energy of edge dislocation near crack, 196 energy of screw dislocation, 172, 174, 176, 179 energy of screw dislocation in whisker, 214 epitaxy, 30 equation of motion of dislocation, 103, 134, 135 equations of motion, nonlinear theory, 252 equilibrium equations, nonlinear theory, 253 equivalence of contour integral and algebraic solutions in anisotropic elasticity, 150 Eshelby, Frank and Nabarro, formula of, 72 Eulerian incompatibility tensor, 249 Eulerian strain tensor, 229, 231 extended dislocation, 45 flux density, 94 flux-line lattice, 335 force on dislocation, 64, 74 forces between dislocations, 66 Frank partials, 45 Frank-Read source, 77 Gebbia's theorem, 169 glide, 6, 55 glide, critical stress for, 77 glide of edge dislocation, 161 glide of screw dislocation, 162 glide plane, 56 grain boundaries, 68 Green's deformation tensor, 229, 233, 236 Green's tensor, 46 Green's tensor for anisotropic materials, 320 half-width of dislocation, 76 helical dislocation, 59 helical dislocation, climb of, 125 Hooke's law, 146 hydrodynamic analogy, 180 hydrostatic pressure, 51 hydrostatic pressure, relation to dilatational strain, 160 image field, 169, 192 image field, geometrical, 198 image force, 188,216 imperfect dislocations, 8, 10 impurities, effect on moving dislocations, 119 incompatibility, relationship to dislocation density, nonlinear theory, 297 incompatibility tensor, nonlinear theory, 248 inductance, (dislocation) mutual, 92 inductance, (dislocation) self-, 92 interaction of dislocations, 66, 91, 161, 163 interaction with elastic inhomogeneity, 170

internal friction, theory of, 116 ionic solids, 27 irreducible integrity basis, 260 jogs, 62 kink motion, 115 kinks, 21, 62 Kroner's incompatibility tensor, 87, 90 Kroner's stress function, 90 Lagrangian incompatibility tensor, 248 Lagrangian strain tensor, 229, 231 line bundles, 337 line tension approximation, 78, 154 line tension of dislocation, 54 linear hardening (stage II), 24 liquid crystals, 16, 28 mapping function, 179 material symmetry, effect on elastic constants, nonlinear theory, 260 metric coefficients, 232, 235 microscopy, transmission electron (TEM), 26 mixed dislocation, 40 molecular crystals, 14, 28 motion, conservative, 55 motion, non-conservative, 58 networks, 24, 41, 68 neutron stars, 29 nodes, 7, 41, 68 parabolic hardening (stage III), 25 partial dislocations, 44 partial dislocations, inelastic forces on, 74 Peierls model, 75 Peierls-Nabarro force, 74 perfect dislocations, 9, 10, 44n photoelasticity investigations, 132 pile-ups, 70, 161, 162 pile-ups, stress field of, 134 pinning, 160 Piola-Kirchoff stress tensor, 251, 253 plane interface, 193 plastic deformation, 124, 131 plastic deformation, rate of, 95 plastic deformation, start of, 65 plastic deformation and dislocation motion, 59 plastic strain, 44, 60 point defects, interaction with, 116 polarization tensor, 89, 100 po1ycrystals, 25 polygonization, 68

polygonization in anisotropic crystals, 161 polymer structures, 335 potential of screw dislocation, 178 prismatic dislocations, 57 pnsmatIc disloc-~tions,fOrces on, 66 prismatic dislocations, formation of, 128 prismatic dislocations, growth of, 128 prismatic dislocations, self-energy of, 94 radiation by moving dislocations, 102, 137 radiation damage, 31 radius of core, 20, 53 rare gases, 28 Riemann-Christoffel tensor, 245 rotation dislocations, 5, 6, 9, 11, 13, 14, 17 screw dislocation, interaction with elastic inhomogeneity, 170,219 screw dislocation in circular cylinder, 171 screw dislocation in general cylinder, 177 screw dislocation in plate, 181, 204 screw dislocation in stress-free rectangle, 183 screw dislocation normal to free surface, 201 screw dislocations, 40 screw dislocations, climb of, 125 screw dislocations, field of, 51 screw dislocations, glide of, 162 self-energy of dislocation loop, 92 Shockley partials, 45 simplification of formulae in anisotropic elasticity, 155 singular points, 16 singularities of other order parameters, 18 slip symmetry, 156 slip systems, 163 small deformation superposed on finite deformation, 276 soliton model, 20 Somigliana dislocation, 170 sound emission, 137 spirals, 80 stacking faults, 44 strain, anti-plane, 171 strain, axes of, nonlinear theory, 238 strain, plane, 183 strain field of edge dislocation, 49 strain field of screw dislocation, 51 strain invariants, nonlinear theory, 238 strain tensor, 42, 48, 87, 146 strain tensor, Eulerian, 229, 231 strain tensor, Lagrangian, 229, 231 strain tensor of edge dislocation, 51 strain tensors, nonlinear theory, 236 stress field, direct observation of, 132

34Y

j)U

Subject index

stress field of dislocation loop, 147 stress field of edge dislocation, 49, 158 stress field of moving dislocations, 101 stress field of screw dislocation, 51 stress field of straight dislocation segment, 149 stress field on dislocation, 62 stress function, 154 stress function, Kroner's, 90 stress functions for edge dislocations, 186 stress tensor, 41, 101, 146 stress tensor, convected, 252 stress tensor of edge dislocation, 51 stress vector, 249 string model, 114 successive approximations, method of, 267, 316 surface tension, 74 surface traction, 169 symmetry transformations, 260 tensor analysis in terms of natural base vectors, 324 tilt boundary, 69 torsion tensor, nonlinear theory, 300 translation dislocations, 5, 6, 12

transmission electron microscopy (TEM), 26 transmission electron microscopy, measurement of images in, 163 twinning dislocations, 81 twinning dislocations, forces on, 83 twinning dislocations, sound emission from, 137 twins, 81 twins, dynamics of, 136 twist boundary, 69 vacancy diffusion, 123 vibrations due to dislocation, 136 Volterra dislocations, 3 Volterra dislocations, nonlinear theory, 263 Volterra process, 3 volume changes caused by dislocations, 288 vortex analogy, 180 walls of dislocations, 67, 68, 161 wedge dislocation, 195 whiskers, twisting of, 212 Zener's formula for volume change produced by dislocation, 290