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. <5 - to stick along the lips S', S" and remove the external stresses applied on S', Sf' to produce the displacement D(r). a
f3 - to displace the two lips
0'
Fig. 2
J$lJJL L
S"
v (a)
Equivalence of rotation (D, Ox) with rotation (D, 0' x') plus translation d.
By construction, D is a constant along the line L. In general, one knows that this is the sum of a translation b and of a rotation Q by an angle Q around an axis Ox. One can therefore have either rotation or translation dislocations or dislocations of a mixed character. It is useful to remember (fig. 2) that a rotation Q(Q, Ox) is equivalent to the sum of a rotation (Q, O'x') by the same amount around a new axis O'x' parallel to Ox and of a translation d, with (1)
(b)
Fig. 1 Volterra's process: (a) in perspective, (b) in section.
According to the self-evident Weingartern theorem, the, relative displacement without distortion ensures that, at the end of the process, internal distortions are continuous across the cut surface. Thus they can be (and indeed are) singular only along the line L. Such dislocation lines are thus characterized by: their geometrical position L. - their infinitesimal strength D. For the media considered here, the end result of Volterra's process is independent of the position of the cut surface S, if this does not touch the outside surface L of the medium. If the cut surface S extends to an outside free surface L, the final result of Volterra's process is still independent of the position of the cut, except for a total rotation of the medium, which does not alter the state of internal stresses. This result can be checked directly: it appears more obvious however with the use of the Burgers circuit (cf. below). In the definition of a relative displacement D one has to define which lip is called S' and which S". This is best done by fixing a direction along the line L of fig. 1, and deciding that, for instance, S' is on the left and S" on the right for an observer lying on the line L along that direction, and looking towards the cut surface. A reversal of the direction along L corresponds to a change of sign for D.
if I is the distance between the two axes of rotation.. For any mixed (translation + rotation) dislocation, one can choose the position of the axis Ox of rotation so that the translation b is parallel to Ox. Some further properties need be stressed; - Any distortion of the lip S' or S" during its displacement can be analyzed as due to a succession of displacements relative to the undistorted positions considered in Volterra's process (fig. 3). Such a distortion can therefore be thought of as adding to Volterra's simple loop L a succession of other dislocation loops l, l' along S' and S". As recalled earlier, any state of internal stress can be relieved by a suitable choice of internal cuts with distortion of the lips. Thus conversely, starting from a strain-free medium, any state of internal stresses and strains can be obtained by a succession of Volterra processes with distortion of the lips of the cuts; it can then be analyzed as due to a collection of dislocation lines.
L
Fig. 3 Distortion of lip S' of the cut in Volterra's process.
J. Friedel
Dislocations - an introduction
- A rotation dislocation (L, Q) is equivalent to a continuous distribution of infinitesimal translation dislocations. This is equivalent to stating that the empty space (or matter in excess) produced by a relative rotation (fig. 1)can be sliced into a succession of infinitely thin slices of constant thickness (fig. 4). Each slice s corresponds to a dislocation loop t with a translation vector equal to the thickness of the slice. Thus to define any state of internal stresses, one only needs translation dislocations.
This equation summarizes the fact that all three dislocations must have the same axis of rotation, with Q = Q 1 + Q 2 and b = + b 2 • At the triple nodes N, N', the dislocations involved then fulfil condition (2) if the directions of circulation are defined as in fig. 6a. More generally, when a number of dislocations meet at a node, their displacements fulfil the condition
6
7
(3) if the directions along dislocations are taken as all pointing towards the node (or all away from the node). This again means that all dislocations must have concurrent axes of rotation, with = 0 and b, = O. For example, the theorem recalled by eq, (1) shows that a pure rotation dislocation L (Q, Ox) can shift its axis of rotation Ox by I (dislocation L 1 ) if it emits a translation dislocation with a translation d given by eq. (1).
Fig. 4 Equivalence between a rotation dislocation L and a continuous distribution of infinitesimal translation dislocations I.
"
~ L1
1
rr
S'
1
S'
l
L
(a)
(a)
(b)
Fig. 5 Motion of a dislocation from L to L': (a) in perspective, (b) in section (conservative motion).
- If the cut surface 8 is extended by d S' (fig. 5), the dislocation line is displaced from L to L'. This usually increases the volume of empty space or of matter in excess, to be compensated for in the Volterra process. The only case where this would not hold would be if the displacement D were a pure translation b, and L were displaced along d8 parallel to b. Such a displacement of a translation dislocation parallel to its translation vector is called a conservative motion or glide. Any other motion is nonconservative and is called climb: it necessarily involves the creation of void or matter in excess, which can be compensated by diffusion only at high temperatures and for slow motion. Any cut surface 8 can branch into two other cut surfaces 8 1 and 8 2 without discontinuity of distortions along the branching line l, if the corresponding relative displacements D 1 and D 2 are such that (fig. 6): D
=
D1
+
D2.
(2)
Fig. 6(a), (b)
S"
S" 2
L 2
(b)
Branching of a cut surface S into two, S l ' S2'
As the distortions are singular along
there is in the immediate neighbourhood of
L a region where they are too large to be analyzed with linear elasticity. A specific
analysis, which will be considered in sect. 2, must be made in that range. Outside this core region, the strains produced by L are elastic, thus directly proportional to D for a given L. The elastic energy stored outside the core region is thus proportional to D 2 . As Li (DJ2 < (Li DJ2, it is clear that energy is gained by splitting a dislocation loop into anumber of loops of smaller displacements. This splitting can be spontaneous and progressive, by the nucleation and progression of nodes as described above, if the dislocations are allowed to move by glide or climb. The classical media discussed here have therefore a spontaneous tendency to store internal strains in the form of a continuous distribution of (translation) dislocations of infinitesimal strength. Then the linear elasticity approximation used here extends all the way to a very small core.
/ 8
J. Friedel
Dislocations
- The equivalency of fig. 4 shows that a rotation dislocation has a large energy compared with a translation dislocation, so that it is usually unlikely to be formed. More precisely, if the position L of the line is at a distance I from the axis of rotation, fig. 4 shows that the core energy is equivalent to the energy of a translation dislocation of translation about Q x I; this is very large if L is not practically along the axis of rotation Ox (l c::::: 0). Furthermore, the long-range distortion energy is impossibly large, in macroscopic tridimensional media, if parallel rotation dislocations of equal strength and opposite signs are not coupled at fairly short distances: otherwise, the translation dislocations I associated with the rotation dislocation L (fig. 4) build very large strains indeed. Using the theorem of eq. (1) we can then state that a pair of parallel rotation dislocations (Q, L), (- Q, L') is equivalent at long range to a translation dislocation of translation d related by eq. (1) to the distance I between the two dislocations L, L'. Only pairs at short distances I have energies which are not prohibitive. Furthermore the motion of such a dislocation pair would involve the emission by one dislocation and absorption by the other of a stream of translation dislocations (fig. 6) which would require a very large expense of energy: such a pair can be considered as "sessile". The corresponding cut surfaces are shown (fig. 7) for when the initial cut is along the two dislocations. S'
an introduction
9
necessarily bounding a surface S ofmisfit or grain boundary, where the anisotropy of the medium has a discontinuity. For perfect dislocations, anisotropy is continuous along the cut surface. D must necessarily be any translation b or a rotation Q that preserves the symmetry of the medium or a combination of the two. In the most anisotropic elastic medium, the possible rotations are all the rotations of multiples of ti around any axis. Less anisotropic elastic media can have axes of rotation of tn, 2n/3, ... or axes of revolution. The corresponding rotation dislocations are allowed, and their multiples. Owing to anisotropy, some rotation dislocations are thus forbidden, and some allowed rotation dislocations are quantized. Furthermore, and contrary to the isotropic case, the end result of a Volterra process for a quantized rotation dislocation is now dependent on the position of the cut surface. Take for instance a medium with cylindrical symmetry (and, as in all elastic classical media, a centre of symmetry). This can be visualized for instance as a frozen liquid crystal in the nematic mesomorphic phase. The specially allowed rotations are all rotations of multiples of 7[ around an axis perpendicular to the axis of revolution, and all rotations around the axis of revolution. Figure 8b, c, d pictures three con-
s
d
I c
t 1
/
t
S" Fig.7
/
1
1/
Cut surfaces S', SI/ associated with a pair of parallel and opposite rotation dislocations (U, L),
~
(-D, L').
L
Finally one can remark that two translations are commutable, but two rotations of finite strength are not in general commutable: exceptions are if the two rotations have the same axis, or if they have parallel axes and are opposite in strength. As a result, there is no difficulty in defining the rotation dislocations of a pair such as in fig. 7 when they have finite strength ± Q, nor in defining any configuration involving a finite number of translation dislocations or of rotation dislocation pairs of finite strength, so long as they do not cut each other.
/
/
-b
(a)
1.2. Volterra's process in a continuous, homogeneous but anisotropic solid medium. Perfect and imperfect dislocations The anisotropy appears in the tensor of elastic constants. In un strained homogeneous media, this tensor has the same value at all points. If the medium is anisotropic, there will obviously be a discontinuity along the cut surface at the end of the Volterra process, if D is not a symmetry operation of the medium. Such general dislocations are usually called imperfect or partial; they are
(b)
(c)
(d)
Fig. 8(b), (c), (d) Three configurations of an anisotropic medium with cylindrical symmetry around a rotation dislocation 2n normal to the axis of revolution, depending on the initial position of the cut surface (a).
J. Friedel
10
Dislocations
11
an introduction
figurations for a rotation of 2n around an axis perpendicular to the axis of revolution, depending on the initial position of the cut surface (fig. 8a). The lines are tangent to the local direction of the axis of revolution. As a result, a pair of parallel and opposite rotation dislocations can have different configurations, depending on their orientation with respect to the anisotropy. Figure 9 shows two configurations of high symmetry for ± tt dislocations of the same nature in the same medium.
(a)
(b)
Fig. 10 Dislocations in crystals: (a) translation dislocation, (b) rotation dislocation.
-B (a)
(b)
Fig. 9(a), (b) Two configurations of high symmetry for a pair of parallel and opposite (±n) dislocations, with axis normal to the axis of revolution in a medium with cylindrical symmetry. (a)
It can also be noted that such quantized rotation dislocations are only equivalent to continuous distributions of translation dislocations if it is accepted that a smooth lip such as S" (fig. 3) takes the average direction of the more rugged lip SI, thus rotates by Q with respect to the direction of the layers on SI.
Fig. 11 Examples of imperfect dislocations in a crystal: (a) translation dislocation with stacking fault F, (b) rotation dislocation with grain boundary B.
1.3. Volterra's process for crystals Here the core structure extends at least over an interatomic distance. The previous analysis extends to crystals by stating that, for a perfect dislocation, D must be an operation of symmetry of the crystalline structure and thus quantized: it can be a translation h equal to a period of the Bravais lattice (fig. lOa). it can be a rotation or a translation-rotation allowed by the symmetry of the crystal (fig. lOb). it can be a multiple or combination of these two types of displacement. As in anisotropic continuous media, the configuration of the crystal around a rotation dislocation depends on the initial orientation of the cut surface. Imperfect translation dislocations border stacking faults (fig. 11a) : imperfect rotation dislocations border (rotational) grain boundaries (fig. 11b). All other properties mentioned in the previous cases extend directly to crystals. Owing to energy considerations, and in conditions where perfect dislocations are reasonably mobile, one expects multiple dislocations to decompose spontaneously
- ---e
Fig. 12 Splitting of a translation b dislocation into a pair of rotation dislocations ±~n (L and L').
into their elementary components, corresponding to the shorter periods of the Bravais lattice and the elementary rotations or rotation-translations allowed by the lattice. The rotation dislocations can only occur in neighbouring pairs of parallel and opposite dislocations. Figure 12 gives an example in a hexagonal lattice. This can be considered as a special splitting of a straight translation dislocation into two rotation ones. Splitting of a perfect translation dislocation into partial translation dislocations bordering a ribbon of stacking fault occurs over more than a lattice period if the
13
J. Friedel
Dislocations - an introduction
stacking fault has especially low energy. Grain boundaries can have a free end as in fig. l lb, only if they correspond to a small rotation (subgrain boundaries). Largeangle grain boundaries meet each other nearly perfectly, so that only reduced internal strains are produced near their triple points.
(ii) If - D is a pure rotation, it varies with the position of C in the way expected if C
12
encloses a varying amount of a continuous distribution of infinitesimal translation dislocations, as defined in fig. 4. 1.5. Extension of the notion of Burgers circuit for rotation dislocations. Disclinations
1.4.
R."~"""",,
circuit for elastic solids
Another way of characterizing the line singularities of the internal stresses and strains in elastic solids is to draw a closed circuit C around the line in the strained solid, and to draw the corresponding circuit in the corresponding unstrained solid. This is possible because at long range, outside the core, the elastic displacement u(r) is well defined and varies continuously in space. '-
M'
S
The notion of Burgers vector is particularly well adapted for the translation dislocations for which it was invented. It is not sufficient to define rotation dislocations in anisotropic solids, whether continuous or crystalline, for it does not distinguish configurations such as those of figs. 8b, c, d, which depend on the orientation of the cut surface. An extension better adapted to rotation dislocations is to follow by continuity the orientation of a local trihedron P xyz which at each point P of C, in the unstrained solid, corresponds to a local trihedron of constant orientation on C in the strained solid (fig. 14). y
S' M
c
(a)
Fig. 13
(b)
(a)
Burgers circuit M M': (a) distorted solid, (b) undistorted solid.
Fig. 14
In the general case, the Burgers circuit in the undistorted solid will not be closed, but the displacement - D will have to be added to close it (fig. 13). This is connected with the fact that u(r) is multivalued, and increases by - D each time C goes once around L:
1~ du
-D.
(4)
This relation is obviously independent of the choice of the origin M on C. This shows clearly that the nature of the line singularity L can be defined as independent of the cut surface S in isotropic solids. In anisotropic ones, the choice of M fixes the relative orientation of D in the unstrained solid: the choice of the cut surface matters for rotation dislocations. We can also note that the sign of D is related to the direction on C, which can be defined unambiguously if a direction of circulation is chosen on L. A reversal in the directions of circulation on L, and thus C, again changes the sign of D. (i) If - D is a pure translation - b, this is a constant whatever the position of C, if it turns once around L. It is the Burgers vector of the translation dislocation.
(b)
Extension of the notion of Burgers circuit for rotation dislocations: (a) strained solid, (b) unstrained solid.
If C does not enclose a rotation dislocation, the total rotation P from M to M' is P xyz turns continuously by U zero. If C encloses a rotation dislocation of rotation from M to M' along C in the unstrained solid (fig. 14b). Again, this amount of rotation is independent of the origin on C (fig. 14a), and thus of the position of the cut surface: it is also independent of the position of C, if it turns once around L; it changes sign with the sign of circulation around C, and thus around L. Finally, if P xyz is defined on M by the local principal directions of anisotropy, we see that different orientations of P xyz on a given point M do correspond to different configurations, as shown for instance in fig. 8. Let (J) be the rotation of the trihedron P xyz from M to P along C, in the undistorted solid (fig. 14b). If C does not enclose a rotation dislocation, OJ describes a closed loop y when P describes C from M to M' (fig. 15a). It C' does enclose a rotation dislocation U, OJ describes an open loop y' of length - U. The possible values of U are distributed as described above, either in a continuous way or along spheres of various radii. The origin of y fixes the choice of cut surface. The description in rotation space (fig. ISb) neglects the translation part of the dislocation, and in particular defines the direction of the rotation axis but not its XYZ
14
J. Friedel
Dislocations - an introduction
position. To follow preceding uses, it might be useful to call that part of a rotation dislocation defined by an extended Burgers circuit disclination. The complementary description of the translation part is given by the Burgers vector in real space (fig. 15a).
- dislocations of the molecular orientation, defined with the use of an extended (rotational) Burgers circuit (fig. 17b).
~ ~
15
/ . // I II ~
-4-,.-/
~
(a)
(b)
~ _~~:I f 1
--{L
I
-.-
-.-
-+- -e-
\ -0-
\
-.- --.-
--.----.-
\ \-0-
-+-\
-.-
Fig. 15 Extended Burgers circuit: (a) real space, (b) rotation space. (a)
In all the solids described so far; the use of this extended Burgers circuit does not add any new information, because, as stated several times, rotation dislocations are equivalent to a distribution of translation ones. Thus except for differences such as those described in fig. 8, the use of normal Burgers circuits is sufficient. This is not so in other media, such as liquid crystals and perhaps some molecular crystals. 1.6. Possible dislocations in some molecular crystals In some molecular crystals, there exists a "plastic" phase just below the melting point where the component molecules keep a crystalline order of their centres of gravity but have a directional disorder at long range. It is just possible to imagine that, below such a phase, there might occur in some cases a phase where the molecules have directional long-range order which has a low energy of anisotropy: configurations such as figs. 16a, b, c, have similar energies. It is clear that, in such a case, there are two possible types of rotation dislocations, which are not necessarily coupled, as assumed so far: - dislocations of the lattice (fig. 17a), defined with the use of an ordinary Burgers circuit
o 00 o 00 o00 (a)
(b)
(c)
Fig. l6(a), (b), (c) Molecular crystal with a low energy of anisotropy in its directional order.
\
-.- -.- ---.- \ ~ ~ ~ ...... ~~\\~ -.-~~,,\
\
"
(b)
Fig. 17 Two types of rotation (1[) dislocations in molecular crystals with uncoupled lattice and directional ordering: (a) lattice dislocation, (b) directional dislocation (lines along the axes of the molecules).
The first type of dislocation can also be obtained by a modified Volterra process which acts only on the positions of the centres of gravity of the molecules without altering their orientation. The second type of dislocation can be obtained by another extension of the Volterra process, where along the cut surface S one rotates the molecules on each of the cut surfaces Sf S" with respect to each other without moving their centre of gravity [de Gennes]. The increase in the number of possible independent singularities is clearly related to the increased number of assumed independent parameters defining the crystalline order: in this case the positions of the lattice nodes and the directions ofthe molecules. Hence the notion of an extended Burgers circuit, which has components in the real translational space and in the space of rotations [Kleman]. We shall discuss as an example a simple case of molecules with an axis of revolution. A change in configuration of the directional order does not involve any shift of the molecules: it can a priori take place much more quickly than a change in the configuration ofthe centres ofgravity. As a result, a singularity produced in such a configuration of directional order by a Volterra process which is not stable energetically will spontaneously evolve towards a more favorable configuration. Two types of such evolutions can be envisaged: (i) Multiple rotation dislocations can split into elementary rotation dislocations" by the building and shifting of nodes. This is similar to but faster than the processes invoked for solids. (ii) Rotation dislocations of multiples of 2n can disappear as line singularities, by the molecules escaping into the third dimension when they are nearing the dislocation core. A Volterra 2n dislocation in this relaxed state is pictured in fig. 18, in projection normal to the line and in section. The sign means a molecule in the plane of figure; the sign til means a molecule normal to the plane of figure; and the sign r- means a
J. Friedel
16
Dislocations
molecule pointing to the right and towards the observer. It is intuitive that this escape into the third dimension might lower the energy for some elongations of the molecules. Thus such dislocations are not only topologically unstable in the sense that a small local distortion can suppress the line singularity: they might (or might not) be energetically unstable as well [Rault; Cladis and Kleman]. Cross-overs between the two regimes are expected for critical elongations of the molecules: singular lines are preferred to singular points for long molecules and thus have a large flexion elastic coefficient. L
<,
I
\
<,
~
/
I
\
<,
I
<,
---
<,
I -:
->
-,
\
I
/
/
/
--
/
N
/
I
\
/-
/'
-:
I
\ <.
-,
\
I
/
L·
~
~
-:
/ /
I
<,
-.
\
I
/ /
-L
Y
<,
/
/
I
\ \
<,
<,
............
-:
/
I
\ <.
-.
<
T
l-
I-
y
-I 1
/
-.
(b)
(a)
Fig. l8(a), (b)
<,
2n dislocation of the directional order, in a relaxed state.
As the escape can a priori occur towards the top or the bottom of fig. l8b, one conceives the possibility of two types of singular points, with a radial orientation around L as shown in fig. 19, at points Nand S [de Gennes]. Such a singular point is characterized by the topology of orientations of the molecules on a two-dimensional closed surface containing the point. Without going into the details of this analysis, one sees that such a closed surface plays for the singular point the role of an extended (rotational) Burgers circuit [Feldtkeller]. It can be checked easily that rotation dislocations of strength ± TC are on the contrary topologically stable: they cannot vanish by a small distortion near the core, and no independent singular points are connected with them [Bouligand]. 1.7. Singularities in
/
-:
T
.:
»>
1/
<
/
an introduction
crystals
It is not intended to discuss this topic fully here, since it is dealt with elsewhere in this
book, but only to stress some differences between liquid and solid crystals. Liquid crystals are anisotropic liquids built up with elongated and quasi-symmetrical molecules: - Nematics are three-dimensional liquids with an axis of revolution parallel to the molecules. - Cholesterics are three-dimensional liquids with a helical axis of rotation + translation normal to the molecules. - Smectics are two-dimensional liquids, i.e. layered structures where each layer, with molecules often normal to the layer (smectic A), has a fairly constant thickness and a two-dimensional liquid behavior.
I -:
->
.............
---
17
s ,,/"
....--
I
/
/
I
-,
/
/
I
\
<,
-.
L
Fig. 19 Two types of singular points of opposite signs Nand S in a perturbed directional order.
They are media with a possible directional order, but no (full) order in the position of the molecules. The distortions in this directional order can store an energy which can, in the limit of small distortions, be described in terms of generalized elastic constants. The situation is very similar to the distortions in direction for molecular crystals (fig. l7b). (i) All possible individual line singularities can be defined starting from a Volterra process applied as if the medium were solid and then allowing for the liquid possible un quantized translation and rotation motions which can relax some of the strains. This relaxation disperses and eliminates all singularities which are not quantized and reduces the energy stored in the remaining quantized ones. It allows some of these dislocations to move and bend, keeping their axes of rotation in their cores. In some cases, it also allows a rotation dislocation to rotate its axis of rotation. All these motions can be visualized as the emission or absorption of continuous distributions of infinitesimal (un quantized) translation and rotation dislocations. As a result rotation dislocations can exist fairly isolated from each other and move freely in liquid crystals. In the simplest case of nematics taken as an example, all translation dislocations and the rotation dislocations with an axis of rotation parallel to the molecules are
18
J. Friedel
eliminated. There only remain the quantized rotation dislocations, of multiples of tt with axes normal to the axis of revolution of the nematic structure. As in the molecular crystals, a further relaxation of the cores suppresses the line singularity, by an escape of the molecules into the third dimension, if the rotation is an even multiple of ti : only singular points remain in that case [Nabarro]. For odd multiples of n, the rotation dislocations have energies which increase only logarithmically with the distance between opposite dislocations, as for translation dislocations in solids. Singular points and lines move freely in the nematic. (ii) As all the translation component is relaxed, the remaining rotation dislocations are most easily characterized by the use of the rotational Burgers circuit discussed for molecular crystals (fig. 15) or of the modified Volterra process introduced by de Gennes. The same is true for some of the dislocations or other equivalent line singularities (focal conics) in the other mesomorphic phases ofliquid crystals. Some other dislocations of these media, which involve at least in part a translation, and the general connections with similar arrangements in solids, need the use of a fully extended (translational + rotational) Burgers circuit. Some of these latter problems are more easily understood in terms of a Volterra process, which must however be used with care in the more complex cases. 1.8. Singularities of other order parameters [Toulouse and Kleman]
Dislocations
19
an introduction
saturated with dislocations; it is however better grounded in second-order transitions, if only because melting is a first-order transition. Consider as an example a classical ferromagnetic crystal, with dimensionality d and n magnetic degrees of freedom per lattice site: d = 1 for a line, 2 for a plane, 3 for a volume; n = 1for the Ising coupling, 2 for the xy coupling, 3 for the Heisenberg coupling. It can be directly checked from the discussion above that the topologically stable defects are respectively points p and line 1 for d = 3 and n = 2 and 3 respectively (fig. 20). The results can be obviously extended to plane singularities P for d = 3, n = I; and, for d :1= 3, the point, line and plane singularities fall on straight lines with a slope one in the (n, d) diagram, a general topological result which can be directly checked. d
4 0
2
3
4
5
2
The study of elastic singularities in crystals and liquid crystals has recently stimulated the general topological analysis of possible singularities of order parameters. A few points might be stressed here: Each type of singularity can be defined using an extended Burgers circuit. - General theorems allow one to state the types of topologically stable singularities. An energy analysis, using the full equation of state for the order parameter, would be necessary to show whether topologically unstable singularities are also energetically unstable. This is however usually to be expected, as the core part of a singularity is generally a region of large stored energy, and it pays to reduce its extent (an obvious counterexample is the hyperbolic branch of a couple of focal conics in an A smectic, which is topologically unstable but energetically stable at least over part of its length: fig. 8 also shows that topologically equivalent configurations need not have equal energies). Singularities ofthe order parameter are expected to be unstable at low temperatures, except possibly for quantum effects in quantum solids (cf. below). Because of their positional and vibrational entropy, one expects them to be thermally excited. Because of their large internal energy, the concentrations in thermal equilibrium usually remain small until, possibly, a critical temperature above which they appear in greater numbers. The collective character of such a possible transition is due to the fact that these singularities interact at long range in such a way as to reduce their internal energy. Thus the singularities of the order parameter might play a specific role in describing the short-range order above the critical temperature where order disappears. This is a suggestion inspired by the old description of a liquid as a crystal
o
-,
Fig. 20 (n, d) diagram for ordered media with a vector order parameter. The hatched region has no possible order at finite temperature.
Thus for d = 3, magnetic walls are singular planes only in the Ising model. In the xy or Heisenberg models, they are regions of continuous rotation of the magnetic moments, only stabilized by the energy of crystalline anisotropy. In the xy model, which forces the magnetic moments to lie in a given xy plane, these walls can contain line singularities, along which the rotation in the wall changes its sign (Neel lines). In the Heisenberg model, the magnetic moments can escape in the third dimension, so that only singular points are stable (Bloch points). For d = 2, n = 2, there is a singular point in the region of allowed order; it is known that although the order temperature is zero, there is a finite temperature above which singular points (vortices) appear in great number. This would be a case where the catastrophic appearance of the singularity is not related to the ordering temperature. Strong analogies exist between quantized dislocations in crystals and liquid crystals and vortex lines in suprafluids or supraconductors. Analogies also exist between the continuous distribution of infinitesimal dislocations and the distribution
J. Friedel
Dislocations - an introduction
of vorticity in normal liquids in motion. Indeed one can suggest that the appearance of turbulence under strong perturbations is a cooperative creation of vorticity somewhat akin to the melting of crystals at high enough temperatures.
This problem has analogies with that of a layer of atoms interacting elastically and deposited on a rigid substrate with which they interact by a potential such as V(r) [Frenkel and Kontorowa]. Some aspects of this type of model need stressing. The tine singularity of a complete elastic body is replaced by a continuous distribution of infinitesimal dislocation along P. The width of this distribution defines a core which is more extended the smaller the amplitude of V is compared with the elastic constants of the two half-crystals. The total line tension of the line, its anisotropy and the frictional force against its motion all decrease with an increasing width.
20
2. Cores of translation dislocations in crystals Deviations from linear elasticity occur near the centre of the line singularities, in the core region. Three types of approach have been used successively in this field, which we shall briefly discuss.
21
L
2.1. Core cut-off [Volterra] This assumes the elastic regime to apply down to within a short distance of the dislocation line, and neglects what happens in the core region. With a suitable core cut-off distance roof the order of the Burgers vector if this is of atomic size, this approximation was shown by Frank to give a fair representation of the total energy. Two corrections should be made:
y
deviations from linear elasticity outside the core - energy stored in the quasi-amorphous atomic structure in the core. However for dislocation densities which are not too large, the long-range elastic distortions dominate in the total energy, and these corrections are often not very important. This approximation is much used for crystal dislocations and has been extended to continuous distributions of dislocations in solids. Deviations from linear elasticity have been used to analyze the couplings with phonons. The approximation essentially neglects the crystallographic and chemical aspects of the interatomic bondings which are modified in the core when a dislocation is created or moved. Even taking into account the anisotropy of elastic constants, one cannot usually hope to analyze in this simple model those properties which follow from the directionality of the crystal structure or of the atomic bonding. 2.2. Dislocations as solitons [Dehlinger, Peierls] To analyze the configuration and motion of a dislocation line in a crystallographic glide plane P, Dehlinger, and then Peierls, used a simplified description of the crystal structure, by assuming that the two half-crystals separated by P were two continuous elastic bodies interacting by a potential V varying sinusoidally with the local relative displacement u(r) of the two half-crystals across P: V =
L
A sin (ku
+ ¢)
dzr.
The wavelength I = 2n/k represents the atomic periodicity along P, and the Burgers vector is assumed to have a component A along k.
o (a)
Fig. 21
(c)
Dislocations in a Peierls sinusoidal potential: (a) straight line perpendicular to k, (b) general "straight" dislocation, (c) kink.
For straight lines perpendicular to k., the motion reduces to a one-dimensional problem equivalent to other problems with the same non-linear potential. This general class ofproblems is reminiscent of that of the solitary wave in hydrodynamics, hence the name of soliton now attached to the corresponding extended singularity. Such a straight line has equilibrium positions distant from each other by multiples of }L. The energy humps between these positions can be overcome, and the dislocation moves as a whole if a stress is applied that is large enough to overcome this solid friction. This is the zero-temperature Peierls stress O'p. A line initially not perpendicular to k will move easily under any small applied stress, until a part perpendicular to k has developed; this is then trapped in the "sessile" position until O'p is applied (fig. 2Ia). Indeed a line not perpendicular to k has a tendency to take a somewhat sinusoidal form so as to lie longer along the valleys of energy minima than across the energy hills (fig. 21b). It can then be considered as possessing a succession of kinks which bring it from one valley to the next (fig. 2Ic). At finite temperatures, a dislocation initially along one energy valley can move under a stress less than O'p by the thermal activation of a double kink in which the individual kinks, once formed, separate quickly because of their extremely shortrange interactions.
J. Friedel
22
This type of model has been much used to explain: a the choice of close-packed crystallographic planes as glide planes, by trying to relate these to such planes, which are distant from each other, with particularly small amplitudes A in the potential V, and thus particularly small line tension for the dislocations. f3 the value of the zero-temperature Peierls stress, related to the value of A. y - the temperature variation of the Peierls stress and internal friction processes assumed to be related to the creation and destruction ofdouble kinks under alternating stresses (Bordoni peak). It is clear that a more elaborate description of the core is needed for case a, especially for close-packed structures, and that case f3 is at best a way of fixing the amplitude A of the potential V. The double-kink model looks more promising for case y. However it should, in most cases, be reinterpreted, the hills and dales of fig. 21 representing much more complex core configurations than those assumed in the soliton model. Indeed, if taken at its face value, the Peierls model should take into account also the periodic variation of V perpendicular to k, This would lead to a periodic variation of the dislocation energy in the y direction (fig. 21) and thus to a friction on the glide of kinks. Double kink formation is expected to limit the speed of straight dislocations at low temperatures, and kink motion of the speed of curved dislocations at high temperatures, in the usual case where the first process has the larger activation energy.
2.3. Splitting of the core [Heidenreich and Shockley] In most compact simple structures, and especially in metals, some planes of stacking fault which perturb little the arrangements offirst-nearest neighbors, and thus have low energy if the interatomic forces are not directional. This is true along the (111) planes of f.c.c., the (110) and (100) of b.c.c., the basal and some prismatic and pyramidal planes ofh.c.p. Even in the diamond cubic covalent structures, (111) stacking faults can be produced which respect the angles as well as the lengths of nearest-neighbor bonds. Perfect dislocations can then lower their energy by splitting into partials separated by such stacking faults. Figure 22 schematizes some such cases met in nature. Another type of possible splitting is for a translational dislocation to split into a pair of rotational ones, as schematized in fig. 12. More complex splittings can be thought of in crystals with larger cells. Such splittings are only well defined if the stacking-fault energy is low enough for its equilibrium width to be more than one atomic distance. This case is met in many metals and covalent structures.
~/////.--
(a)
Fig.22
(b)
Three cases of dislocation splitting: (a) glissile, (b) and (c) sessile.
Dislocations
an introduction
23
It is also seen that all but the case of fig. 22a are sessile: the dislocation must recompose to be able to glide as a whole; this requires a temperature high enough for a "constriction" to appear and develop by thermal fluctuation; or exceptionally, it can require a large enough applied stress for recombination to be forced on the whole of the dislocation. In the case a, glide normally occurs in the plane of splitting. Cross slip into another slip plane requires a high temperature for thermally activated cross slip and/or a large enough applied stress. This concept of splitting has been checked directly using transmission electron microscopy. It has helped to understand many aspects of slip in common metals and covalents: choice of slip plane in the cubic and hexagonal structures (fig. 22a), thermally activated cross slip in these phases, more difficult motion of screw than edge in b.c.c. structures, owing to a sessile splitting of the screws (fig. 22c), relation of the amount of splitting with ease of twinning.... It is thus clear that, when wide, such splittings must be taken into account in the study of dislocations. One outstanding question is then the exact value ofthe stackingfault energy, and its variation with composition or with temperature. When not so wide, such splittings can give only a qualitative description of the properties of dislocations. This is for instance the case with metals such as Al or with the ionic solids.
2.4. Atomic description of the core Compact structures with dislocations not widely split or complex structures are obviously better treated by a direct computation of the atomic structure of the core. Even in widely split dislocations, there remains the description of the cores of the partials. Exact computations are practically impossible except in rare gases and possibly simple ionic solids. For metals and covalents, the interatomic interactions are but roughly known and only model computations can be made, often of somewhat doubtful validity. A further difficulty is that one can compute simply only dislocations in equilibrium, in simple straight positions, possibly under stress. One can thus manage to compute positions of lowest energy and something like the zero-temperature Peierls stress: but nothing related to thermally activated processes has been computed so far. Configurations of dislocations pointing in less close-packed directions, in a glide plane, or going out of their glide planes, have not been practically tackled either. It is clear however that a much more aggressive policy in this field would be invaluable, if simple typical situations were fully studied with reasonable model interatomic forces. A general remark can however be made: in covalent structures, the creation and motion of a core, whether for a perfect (unsplit) or for a partial dislocation, will involve the breaking of interatomic bonds. In the Peierls language, one expects nearly equally difficult creation and motion of kinks, thus large Peierls stress at zero temperature and thermally activated glide at high temperature, with an energy of activation of the order of the bonding energy so a difficult glide even though the split disloca-
24
J. Friedel
tions are "glissile" in the diamond cubic structure: one also expects that, owing to this large friction, the partials do not necessarily take their equilibrium distance, or even might keep together in a perfect dislocation on some lengths of the dislocations. This seems indeed the case in these solids. Contrariwise, in metals and ionic solids, with essentially central forces, the motion of a core is not expected to give rise to a large frictional force, because the effects or' some bonds widening while others shorten should more or less compensate. Indeed the Peierls friction on split dislocations which are glissile usually seems to be very small, if at all measurable. Indeed the Bordoni peaks of internal friction, if due to the thermally activated formation or motion of kinks, occur at low temperatures and involve reduced energies. Easy glide of dislocations thus require two types of conditions to be fulfilled together a geometrical condition, on glissile splitting a bonding condition, on motion of the core.
Dislocations
an introduction
25
observation and the characterization of its long-range (pile-up) versus short-range (three-dimensional network) structure and of its anisotropy. - Stage III or parabolic stage: in this stage, the external stresses are large enough to produce a recovery by cross slip which decreases the dislocation density and thus produces a hardening less marked than in stage II. More or less well defined polygonized cell structures are developed. If the beginning of stage II has given rise to many analyses of cross slip and its altogether convincing description, the state of affairs is much less clear once the cell structure is established. No convincing description exists so far of the processes by which it grows and thus produces a parabolic type of hardening. It is probable that one of the processes involved in the growth of the structure is a catastrophic destruction by cross slip of some polygonized walls, which would start on the most stressed dislocation, and then propagate to its neighbors. The same cell structure occurs in medium-temperature creep and its evolution should be studied there too. 3.2. Dislocation networks in recovered f.c.c, metals
3.1. Dislocation networks in strained f.c.c. metals
Studies of heat treatments after these various strainings in f.c.c. metals show better defined polygonized cell structures. At high temperatures, polygonization grows by vacancy diffusion, by mechanisms which are well understood in principle. The same cells occur in high-temperature creep. It is however noteworthy that no very systematic studies of polygonization have been made after well defined deformations in the various stages recalled above, to see in detail the relations between the deformation and polygonization textures.
It is indeed clear that the three stages in the straining curves of f.c.c. single crystals do
3.3. Deformation and recovery in other structures
3. Dislocation ensembles If the properties of individual dislocations are, on the whole, mastered and understood
in the simpler crystalline structures, one is very far indeed from this happy state of affairs for the large dislocation densities involved in straining processes.
correspond to three rather different types of dislocation arrangements: - Stage I or easy glide, where dislocations of one slip system mostly develop. This is a rather inhomogeneous distribution, usually with slip bands parallel to the active (111) plane and containing more dislocations than the surrounding material. Hardening seems connected with kink bands normal to the [11OJ active slip direction and made up of loose pile-ups of dislocations of opposite signs. Clearly this is a difficult stage to describe by a few parameters and to analyze in detail. Stage II, or the linear hardening stage, where secondary slip systems become very active: they build with the primary system sessile Cottrell-Lomer locks on which active dislocations can pile up; stress relaxation of the pile-ups by secondary systems finally leads to a three-dimensional network of increasing density. It has been fairly clearly demonstrated that, in this range, hardening is mainly (but not only) a function of the average density of this three-dimensional network of dislocations. But the fact that the network is not isotropic is demonstrated by experiments where, after straining in one direction, one starts straining in another, possibly even the opposite, direction; one always observes in the second straining an elastic limit lower than that produced in the first one (Bauschinger effect). Much remains to be done on the development of the dislocation network, especially under complex straining conditions, on its
It is somewhat of a surprise to observe that the same three stages of deformation and the same recovery structures are observed in a number of other metallic structures notably the b.c.c. and, with some qualifications, the h.c.p. ones. The same is true for a number of ionic solids. Especially in the b.c.c. metals, it is not clear what are the equivalents of the Cottrell-Lomer locks of stage II or cross slip of stage III in the f.c.c. structure. 3.4. Deformation of polycrystals The relation of the straining of a polycrystal to that of a single crystal is understood in principle, and Kroner's work in this field has much helped to develop a simple and realistic approximation. The study of large-deformation textures has been much helped by the use of the development in spherical harmonics of X-ray or neutron pole figures. It is certainly one field where the points of view of mechanics and metallurgy have been brought very near. Still, the plastic properties of the individual grains are very schematized: and the effect on textures of twinning or of the development of stacking faults is still far from clear. The same is of course true for the relations between textures of deformation and recrystallization.
26
Dislocations - an introduction
J. Friedel
3.5. Inhomogeneities in deformation We include here various topics: - Luders bands. - Thermoplastic instabilities. - Striction and failure and, conversely, superplasticity. _ Alternations of recrystallization and deformation in hot working. _ Plastic deformation in front of a moving crack and the connected problem offatigue. In this field, emphasis in recent years has been on trying to develop macroscopic mechanical models that help to describe satisfactorily the inhomogeneities. This was particularly clear in fracture and fatigue, where the measurement of the plastic friction due to propagation in various materials in various conditions helped to reassure industrialists, and especially those in aeronautics, about the harmlessness of many small cracks produced. The same approach has helped to define the favourable values of coefficients in phenomenological equations of plastic behaviour to obtain superplasticity. It now seems time to reassess from a fundamental point of view the wealth of phenomenological results thus obtained, and to try and understand the elementary processes involved, with a view to developing some systematics.
4. New techniques for old problems Transmission electron microscopy has remained the major tool for the observation of dislocations in motion in crystals. Three significant developments must be stressed:
_ The use of special contrast techniques has allowed one to observe details on a nearly atomic scale. Thus the weak-beam contrast has led to the systematic study of the splitting of dislocation cores in f.c.c. metals and diamond cubic semiconductors, confirming earlier but controversial results in the latter case. The use of two-beam contrasts allows a direct inspection of the regularity of the crystal structpre on the scale of the lattice cells and of the atomic structure of dislocation cores. ~ But an even more important development has been the systematic use of high energy electron microscopy. This technique has its own problems, be it the m~)fe difficult analysis of the many-beam contrasts or the radiation damage produced by the electrons. But it is essential to observe dislocation structures on a scale of a few 100 to 1000 A, as produced in many straining conditions. This technique has confirmed in detail the models of micro and macroyields in b.c.c, metals at low temperatures associated with a low and strongly temperature-dependent mobility of the screw dislocations, as predicted from an analysis of possible core splittings in such lattices. Systematic observations of straining and recovery at higher temperatures seem to give very interesting results on cross slipping and diffusive motions of dislocations. Finally observation in the microscope has been essential to our understanding of the early stages of formation of loops, tetrahedra or voids by radiation damage. _ Very fragile materials are now observed, using mostly low temperatures and short
27
observation times: this refers to ionic solids, polymers, biological materials akin to liquid crystals. The X-ray techniques are also being rapidly extended at the moment: - The Lang technique is extended to the observation of moving defects by the use of new fast detectors and more intense X-ray beams. But really fast moving defects are now observed by a technique more akin to the Guinier-Tennevin set-up, using the very intense white spectrum of cyclotron radiation. - A systematic X-ray analysis of deformation and recrystallization textures has been made possible by the use of a mathematical development in spherical harmonics of the intensity of the X-ray spectra. The neutron techniques have also been developed. - Neutron topography has proved interesting in magnetic materials. - Texture analysis can also be made now using neutrons. - The distortions. around dislocations have been studied in magnetic materials. Ion emission microscopy has mostly been used to try to study the atomic structure of the dislocation cores. This has led so far to results less convincing than for the structure of grain boundaries, mostly because the high electric field present during the observation probably alters the amount of splitting near the surface. In addition the optical microscope has been systematically used to study textures and defects in liquid crystals and related materials. Finally, in the study of the physical properties of dislocations, all the corresponding methods are used. Two recent results are
- The connection in luminescent diamonds between the distribution of luminescent centres and of dislocations, as observed by the Lang technique. The observation by EPR of large magnetic polarons associated with extra carriers trapped on otherwise neutral dislocations in semiconductors, with paramagnetic broken bonds in their core.
5. More complex materials Based on a reasonable understanding of the simple metal oxides and semiconductors which developed in the '50's, one observes in the last ten years a definite trend towards broadening the field of research to include more complex materials. This has brought the classical physical metallurgists in contact with other fields of research, with mutual benefit in cases such as geophysics where the contact has been prolonged and deep. In other fields, one has seen physicists painfully rediscovering well known results ... - The field of ionic solids, and especially oxides, is of direct interest to metallurgists; and it is somewhat surprising that it took so long to develop in full in the dislocation field. It probably has to be realized that the plastic properties are dominated in these materials, as in metals, by the structure and behavior of the core. Systematic studies
28
Dislocations - an introduction
J. Friedel
of the effect of crystal structure, temperature and purity are less advanced in this general field, admittedly more complex than that of metals. Applications to magnetism, geophysics and building materials are, however, obvious. - Classical covalent solids used by the semiconductor industry have been the subject of much technical research. There seems to be now a better contact between electronics and metallurgy laboratories on the characterization and study of dislocations. But the detailed description of the atomic and electronic structures of the dislocation cores and their connection with the way the dislocations were introduced, or with doping and illumination, are still matter of controversy. A more systematic relation of plastic properties with cohesion in semiconductive compounds and alloys would probably be rewarding, as clearly the creation and motion of dislocations involves the breaking of covalent bonds. A definite extension to more exotic materials, such as covalent chain crystals (S, Se, Te) or covalent plane crystals (graphite, arsenic) has been started and clearly shows a preference for dislocations which avoid the breaking of covalent bonds, both for their creation and their motion. In the field of molecular crystals, very little work has taken place so far that could be called systematic. Thus the plastic properties of "plastic" crystals, the defects in the crystallinity of polymers, the interaction of dislocations and domain walls in organic ferroelectrics are three examples of pretty open questions. The related field of liquid crystals has seen an explosive growth in the last years. This is a classical example of discontinuous growth on the research front. Indeed the mesomorphic phases of "thermotropic" liquid crystals were observed by Lehmann at the end ofthe last century and classified by G. Friedel before 1920. Lehmann observed the line singularities in the molecular arrangements of these oriented liquids, and G. Friedel described essentially correctly the molecular arrangements around the ,line and point singularities. These line discontinuities are in most cases equivalent to Volterra's rotation dislocations. They are therefore very clearly the earliest examples of dislocations observed in nature and understood as such. What made this possible was the low viscosity of liquid crystals, which allows fast recovery thus a quick achievement of nearly perfect arrangements, with low densities of dislocations, together with a strong coupling of light with the distortions around these defects, which makes them easily observable in the ordinary microscope. Nothing but a mental block or a lack of impetus from possible applications can then explain why a field which was flourishing in the 1920's lay practically dormant for more than 30 years before getting a new lease of life. This proved to be extremely fruitful on the dislocation side, by extending the notion of line singularities to ordered structure without a lattice, in phases of increasing complexity. Besides direct connections with line singularities in related problems of magnetism, suprafluidity or supraconductivity, work in liquid crystals has been the starting point of a general classification of singularities of ordered structures which is briefly analyzed above. It has been also extended to related structures: "lyotropic" liquid crystal phases of the soap family, and various biological materials (membranes, some chromosomes and even the cellular arrangement of the skin of some crabs !). This rapid survey should also mention work on the crystalline phases of rare gases.
1U'''''JA~'~~
29
rather humdrum and somewhat incomplete work on the heavier gases, one mention at least two recent works on He.
- Work on crystalline He 4 under pressure seems to show conclusively that, at least in the range of temperatures explored, plasticity in this phase is through a classical and hysteretic motion of dislocations. Thus evidence is there against a fast quantum tunnelling of either dislocations or vacancies. - Work on crystalline He 3 again indicates that dislocations can be accumulated by work hardening, and that they can capture vacancies and pin them down. A related problem is that of the possi ble plastic deformation of the crystalline crust of neutron stars. It seems possible that the "starquakes" observed in some pulsars are due to "catastrophic" deformations of the crust. However, by analogy with what is known of other supraconductive b.c.c. materials such as Nb, it seems likely that these are not cracks but localized thermal spikes developed in a poor conductor by the heat released in local deformation bands: this local heating strongly decreases the macroyield stress in the band, thus favouring an accelerated straining within the bands. Finally one should bear in mind that Volterra's initial analysis referred to the internal stresses of a macroscopic continuous solid. Thus the concept of continuous distribution of infinitesimal dislocations has had a rich and rewarding field of application in the analysis of internal elastic stresses produced in various ways: inside and around more or less coherent precipitates; around cracks and near indenters; in polycrystalline or polyphase materials prepared in various ways so that the various grains mutually exert internal stresses; in non-saturated magnetic materials, when a magnetostriction is present; etc ... It is also clear that any grain boundary and some phase boundaries can formally be described as equivalent to a (planar) continuous distribution of infinitesimal dislocations. A relative sliding of the grains parallel to a fairly flat boundary can be thought of as due to an evolution and motion of these dislocations along the boundary. It was also known for a long time that in low-angle boundaries, such a distribution condenses into a discontinuous distribution of lattice dislocations. It is one of the advances of recent years to have shown that in many cases large-angle boundaries can be thought of as a similar superposition of a perfect twinning arrangement and of a small-angle boundary produced by an array of twinning dislocations. These perfect twins are often of the kind first introduced by G. Friedel, i.e. such that the two grains have a common superlattice. Still along the same lines of thought, one can think of a liquid or a glass as a crystal full of dislocations. This is not a particularly helpful description, as these dislocations very strongly interact and thus have not much physical individuality. But this picture helps to understand that a glass can be plastically deformed at low temperatures by the relative shear of two parts along a sliding surface that is macroscopically flat, through the motion of a continuous distribution of supplementary infinitesimal dislocations. If the surface of contact was atomically smooth, this motion could be a pure glide. And, after a suitable amount of slip, some kind of short-range order could be reestablished across the sliding surface, thus minimizing the energy of the "stacking fault" produced along the slipping surface. Because of atomic roughnesses, some irregular climb on an atomic scale is involved, leading to irregular compressions and
30
J. Friedel
tensions across the slipping surface which compensate at large distances and thus do not store much energy. It is indeed the same kind of description which has been used to describe the relative sliding of two crystalsalong a fairly flat boundary.
6. Physical properties other than plasticity Besides general connections with properties of singularities of other ordered phases, such as magnetism, supraconductivity, suprafluidity, one should mention four fields of special interest: - Electronic properties in insulators and semiconductors. The direct connection of crystal dislocations with the transport and optical properties of such materials has been clear for many years, and much effort has been spent in the electronics industry either to eliminate the dislocations for "perfect" crystals or at least to characterize the dislocation contents. General ideas about acceptor and donor states related either to broken bonds in the dislocation core or to the long-range deformation potential have been aired for quite a time. However very little effort has been made so far to study the electronic structure and excitations of dislocations with known geometry and core structure. This is particularly clear in Si or Ge where core splitting is only beginning to be studied; but even in Te, where some systematics ofthis kind have been tried, it is far from clear that the dislocations introduced are as regular and straight as was assumed. The influence of dislocation networks on the hysteresis of magnetism or supraconductivity has been similarly studied with a view to technical applications. But more work would be of interest on the exact nature of the coupling of individual dislocations with the magnetic or supraconductive singularities; also works on the hysteresis due to well defined dislocation arrays are still very few: e.g. screw dislocations produced by macroyield at low temperature in supraconductive Nb or ferromagnetic iron; three-dimensional networks produced by low temperature straining in ferromagnetic nickel; well defined cell structures obtained in stage III of all these metals. It is also clear that the magnetic hysteresis behaviour of strained or polycrystalline ferromagnets has many aspects similar to the plastic hysteresis of crystals, and that the same kind of statistical models probably apply. - Surface properties. The role of emerging dislocations in the now classical fields of crystal growth and thermal or chemical etching could with advantage be reconsidered with modern methods of surface observation.
But a more active field has recently been that of epitaxial layers. It is clear now, as had been suggested theoretically, that the strongly bonded chemisorbed epitaxial layers can adapt perfectly on an atomically smooth crystalline substrate if their thickness is of atomic dimensions. Epitaxial dislocations appear on the interface with the substrate for thick enough layers, so as to relieve the growing strains when as usual the two phases have not exactly the same lattice parameters. Volume dislocations can also appear to relieve further strains due to a difference in thermal expansion of the two phases, if the epitaxial layer is submitted to a change in temperature after being formed; volume dislocations also appear to relieve chemical
Dislocations
an introduction
31
stresses if, owing to heat treatment, interdiffusion can occur across the interface and concentration gradients build up near to it. All these well-known aspects seem only recently to have been taken properly into account by the electronics industry in problems of epitaxial p-n junctions in semiconductors or magnetic bubble deposits. More loosely bonded physisorbed epitaxial layers can show more complex behaviors even at atomic thicknesses. In the typical field of rare gas monolayers on (0001) planes of graphite, epitaxy at low temperature can be exactly coherent or show an approximate epitaxy which preserves the directions of the atomic rows; this is obtained by introducing a grid of epitaxial (translation) dislocations. As temperature increases or the vapour pressure decreases, one seems to reach a quasi-liquid state of the monolayer, which can possibly be described by the cooperative stabilization of (rotation or translation) dislocations piercing the monolayer. The description is still deduced from rather indirect or global observations, and it would indeed be of fundamental interest to observe directly and characterize in more detail the dislocations involved in these various phases. Indeed various aspects are still poorly understood: the thermodynamics of melting; and the conditions of approximate epitaxy (configuration, hysteresis, entropy of vibration). The "melting temperature" of these two-dimensional layers shows some similarity to the temperature of appearance of vortices in ferromagnetic monolayers with moments in the layer (xy model); the passage from approximate to exact epitaxy is analogous to many other cases of passages from incommensurate to commensurate order parameters in threedimensional lattice or magnetic superstructures. - In the field of radiation damage, great efforts have been made to understand the nucleation and development of dislocation networks and/or cavities by elimination ofthe Frenkel defects produced under heavy radiation doses. In the condition oflarge supersaturations which can obtain, it is not clear whether one can use concepts of near equilibrium or, more likely, a purely kinetic description. This certainly applies to the nucleation of dislocation loops versus voids (where gaseous atoms are needed) and the choice of atomic planes for the loops in anisotropic materials. But this remark might possibly apply also to the formation of more or less regular arrays of voids, loops or tetrahedra observed very frequently under heavy doses. The whole field still needs very basic and fundamental experiments to decide between various possible interpretations. This is crucial in understanding the problems of irradiation swelling and irradiation growth met in the materials used in fast nuclear reactors.
References for sect. 1 Bou1igand, Y., J. Physique 35 (1974) 215, 959. Burgers, J. M., Proc. Kon. Ned. Akad. Wet. 42 (1939) 293, 378. Cladis, P. and M. Kleman, J. Physique 33 (1972) 591. de Gennes, P. G., The Physics of Liquid Crystals (Oxford University Press, 1974). Doring, W., J. Appl. Phys. 39 (1968) 1006. Fe1dtkeller, E., Z. angew. Physik 17 (1964) 121; 19 (1965) 530. Frank, F. c., Disc. Farad. Soc. 2 (1933) 883; Phil. Mag. 42 (1951) 809.
32
J. Friedel
CHAPTER 1
Friedel, G., Ann. Physique 18 (1922) 273. Friedel, J., Dislocations (Pergamon Press, London, 1964). Friedel, J. and P. G. de Gennes, Comptes Rendus 268 (1969) 257. Friedel, J. and M. Kleman, J. Physique 30 (1969) C4; Nat. Bur. Stand. (DS) Spec. Publ. 317 (1969) 607. Kleman, M., Adv. Liq. Cryst. 1 (1975) 267. Kleman, M., Points, Lignes, Parois. Ed. de Physique, Orsay (1977). Kleman, M., L. Michel and G. Toulouse, J. Physique 38 (1977) 195. Kroner, E., Kontinuums theorie der Versetzungen (Springer, Berlin, 1958). Lehmann, 0., Fliissige Kristalle (Engelman, Leipzig, 1904). Meyer, R. B., Phil. Mag. 27 (1973) 405. Nabarro, F. R. N., Theory of Crystal Dislocations (Clarendon Press, Oxford, 1967). Nabarro, F. R. N., J. Physique 33 (1972) 1089. Neel, L., J. Phys. Radium 17 (1956) 250. Rault, J., J. Physique 33 (1972) 383. Toulouse, G. and M. Kleman, J. Physique Lettres 37 (1976) L149. Volterra, V., Ann. Ecole Normale Sup. Paris (3),24 (1907) 400. Weingarten, G., R.C. Accad. Lincei (5), 10 (1901) 57.
Crystal Dislocations and the Theory of Elasticity A. M. KOSEVICH Physico-Technical Institute of Low Temperatures Lenin's Prospect 47, Kharkov 86, USSR
References for sects. 2-6 Aerts, E., P. Delavignette and S. Amelinckx, J. Appl. Phys. 33 (1962) 3078. Amelinckx, S. and P. Delavignette, J. Appl. Phys. 44 (1962) 1459. Bordoni, P. G., Ricerca Sci. 19 (1949) 851. Cottrell, A. H., Phil. Mag. 43 (1952) 645. Dehlinger, D., Ann. Physik (5), 2 (1929) 749. Frank, F. c., Proc. Phys. Soc. A62 (1949) 202. Frenkel, J. and T. Kontorowa, J. Phys. Chern. Moscow 1 (1939) 137. Friedel, J., Comments on Solid State Physics 1 (1968) 24; Revista del Nuovo Cimento 1 (1970) 110; J. Less-common Metals 28 (1972) 241; in Physics 50 years later, IDPAP 1972, Nat. Ac. Sci. Washington (1973) 193. Heidenreich, R. D. and W. Shockley, Strength of Solids, Bristol Conf. 1947 (Phys. Soc. London, 1948). Mott, N. F. and F. R. N. Nabarro, Strength of Solids, Bristol Conference 1947 (Physical Society, London, 1948). Peierls, R., Proc. Phys. Soc. 52 (1940) 321. Schoeck, G. and A. Seeger, Acta Met. 7 (1959) 469. Seeger, A., Phil. Mag. 1 (1956) 651. Shockley, W., Trans. AI ME 194 (1952) 829. Shockley, W., Phys. Rev. 91 (195,3) 228.
© North-Holland Publishing Company, 1979
Dislocations in Solids Edited by F. R. N. Nabarro
35
Crystal dislocations and the theory of elasticity 8.3. The effective mass of a dislocation 107 8.4. Dislocation damping in a medium having dispersion of the elastic moduli 8.5. The string model
Contents
114
9. Dislocations and point defects 116 9.1. Elastic interaction of dislocations with point defects 116 9.2. Dynamical retardation of a dislocation by heavy impurities
1. Introduction 37 2. Elementary properties of dislocations 2.1. Edge and screw dislocations
9.3. Speed of climb of an edge dislocation
9.4. The climb of a helical dislocation 125 9.5. The formation and the growth of prismatic dislocation loops
38
38
2.2. A dislocation cannot end within the crystal
Addendum (1976)
40
References
2.3. Dislocation loops and plastic strain 41 2.4. Partial dislocations and stacking faults
44
3. Elastic deformations in the presence of a single dislocation 3.1. The displacement field around a dislocation loop
46
46
3.2. The strain and stress field of the straight edge dislocation
49
3.3. Deformations near a screw dislocation and the symmetry of a medium
51
3.4. The elastic energy of the straight dislocation and the problem of thermodynamical equilibrium 4. The movement of dislocations and plastic deformation of the crystal 4.1. Conservative motion (glide) 55 4.2. Non-conservative motion (climb)
55
58
4.3. The relation of the dislocation motion with plastic deformation
59
5. Interaction of dislocations with an elastic field 62 5.1. The action of a stress field on a dislocation 62 5.2. Forces between straight dislocations
66
5.3. Polygonization and a dislocation model of grain boundaries 5.4. Planar dislocation pile-ups
70
5.5. The Peierls-Nabarro force
74
68
5.6. Dislocation sources 77 5.7. The dislocation model of the twin
81
6. Systems of stationary dislocations in a crystal
85
6.1. Continuous distributions of dislocations
85
6.2. Dislocation polarization of the deformed crystal 88 6.3. Kroner's stress function for an isotropic medium 90 6.4. Interaction between dislocation loops 6.5. The self-energy of a dislocation loop
91 92
7. Dynamics of a crystal with dislocations
94
7.1. The dislocation flux density tensor
94
7.2. The elastic field of moving dislocations
98
7.3. The stress field in the linear approximation of the dislocation velocity 7.4. The radiation field of moving dislocations 102 8. Equation of motion of a dislocation
103
8.1. The field nature of the equation of motion of a dislocation 8.2. The explicit form of the equation of motion of a dislocation
103 104
101
53
138
132
119
121 128
109
1. Introduction
-:
concept of a dislocation in a crystal, together with the different dislocation representations and dislocation models associated with this concept, has been used most extensively to give an interpretation of many different physical phenomena in solids. The great popularity of the dislocation concept among physicists, who discuss "from a unified point of view" a tremendous amount of experimental data, is apparently due to a considerable degree to the fact that the concept of dislocation is very broad, and admits of many, varied, microscopic models of the structure and properties of this singularity in the crystal. The latter makes it very easy to adapt dislocation models in a crystal of any symmetry to the description ofmany phenomena which at first glance do not have a common physical nature. However, the greater number ofthe significant physical properties of the dislocation are not connected with their microscopic models, and can be described phenomenologically within the framework of macroscopic elasticity theory, to the same degree that elasticity theory is capable of describing the propagation of sound waves of not too short a wavelength in a crystalline body. A dislocation theory based on the notion of a continuous medium and founded on elasticity theory is sometimes called the continuum theory of dislocations. This theory has at present been well developed and can be formulated and expounded in the same way as other branches of theoretical physics. In particular, there exists a complete system of equations which relates, for the general case, the elastic fields with the distributions and fluxes of the dislocations. There exist rigorous equations for obtaining the plastic deformation of a body from the known dislocation motion. The equations of motion of the dislocations under the action of specified forces are obtained. The present chapter is an attempt to explain the elementary properties of dislocations and to give an exposition ofthe principles of the continuum theory ofdislocations in the general scheme of elasticity theory. It provides a description of dislocation properties that do not depend on microscopic models. The task facing the author of the present chapter is very hard since there is a fine current monograph by F. R. N. Nabarro on the theory of dislocations [lJ where these problems are discussed fairly completely. However, the author hopes to avoid trivial repetition by the choice of problems to be discussed. It is not our purpose to review the application of dislocation theory to concrete problems of the plasticity and strength of solids since this is covered in many of the following chapters.
38
A. M. Kosevich
Ch.l
2. Elementary properties ofdislocations 2.1. Edge and screw dislocations There exist many and varied microscopic models for dislocations of different types in crystals. The simplest model seems to be that where the dislocation line is the edge of the "extra" half plane inserted into the crystal. Figure 1 shows the most generally used atomic diagram of this model, where the added half-plane is the upper half of the y-z-plane. The edge of the "extra" half-plane (z-axis in fig. 1) is then called the edge dislocation. The dislocation is a linear defect of the lattice. The distortion of a regular crystal structure occurs only in the immediate neighbourhood of some line (the dislocation axis) and the region of the irregularity extending along the dislocation axis has a cross section with dimensions equal to a few lattice periods. If the dislocation is surrounded by a tube with a radius of the order of magnitude of the lattice period, then the crystal outside the tube can be considered as perfect and subjected only to elastic deformations. The crystal planes fit together in an almost regular manner outside this tube whereas inside the tube the atoms are essentially displaced from their equilibrium positions in the perfect crystal and they form here a certain structure known as a dislocation core. In fig. I the atoms of the dislocation core are conventionally situated along the contour of the shaded pentagon.
Fig. 1 A model of the atomic positions near an edge dislocation along the z-axis.
§2.1
Crystal dislocations and the theory of elasticity
the elastic displacement vector II receives a certain finite increment b which is equal to one of the lattice vectors in magnitude and direction. This property may be written as (1)
The direction along the contour is that of a right-hand screw relative to the chosen direction along the dislocation line I, that is, relative to the unit vector t tangent to the dislocation line. The dislocation line itself is a line of singularities of the elastic field.
Fig. 2 A dislocation line I and a right-hand circuit s, which encircles this line.
The vector b is called the Burgers vector of the dislocation. It is evident that possible Burgers vector values are determined by the crystallographic structure of the body considered and they correspond, as a rule, to a finite number of the vectors of the chosen lattice. As regards the dislocation line I, it may be considered in the macroscopic treatment as a smooth curve. The edge dislocation is represented by the line 1along which the vectors t and bare perpendicular. On diagrams it is usually represented by the symbol 1- where the vertical stroke points to the side from which the "extra" half-plane was inserted. In the representation given by fig. 1 the edge dislocations with opposite directions of b differ in that the "extra" crystal half-plane lies above or below the z-x-plane. Such dislocations are said to have opposite signs.
o To see the deformation far from the dislocation, we consider a closed circuit of lattice points which encircles the origin in the x-y-plane, and follow this circuit around the dislocation core. If the displacement of each point from its position in the perfect crystal is denoted by the vector II, the total increment of this vector around the circuit will not be zero but will equal one lattice period in the x-direction. It is this singularity of the dislocation deformation that may be considered as the primary one in the macroscopic definition of the dislocation in the crystal. The majority of essential physical properties of the dislocation are not associated with their microscopic models and may be described phenomenologically within the framework of the elasticity theory on the basis of this definition. Thus, the dislocation in the crystal is a singular line 1having the following properties: after a passage round any closed contour that encircles the dislocation line 1 (fig. 2),
39
0' Fig. 3 A part of the crystal with a screw dislocation along the line 00'.
A. M. Kosevich
40
Crystal dislocations and the theory of elasticity
Ch.1
41
If the vectors t and b are parallel, then the corresponding dislocation is called a screw dislocation. The presence of a straight screw dislocation in the crystal transforms the crystal planes into a helicoidal surface, Figure 3 shows the arrangement of atomic planes for the case of a screw dislocation whose axis coincides with the line 00'.
D[:001)°4
Fig. 5 A dislocation line I which ends in the medium at P. The end of the dislocation is surrounded by a cap 17 spanning the contour s.
8
I
/ //
V/ c
Fig. 4 A dislocation loop consisting of a screw type segment AB, a mixed type segment BC, an edge type segment CD and a segment of helical dislocation DA.
If the dislocation segment is neither perpendicular nor parallel to the Burgers vector, it is called a mixed type segment. The dislocation segments of edge, screw and mixed types may lie continuously along any curve forming a single dislocation line or a dislocation loop. Figure 4 shows schematically the dislocation loop ABCDA consisting of a screw type segment AB, a mixed type segment Be (450 dislocation), an edge type segment CD and the segment of the helical dislocation DA.
Let us assume that the dislocation line I 1 with the Burgers vector b, divides at a point A (fig. 6) into two dislocation lines I 2 and I 3 with the Burgers vectors h2 and h 3 , respectively. In this case the dislocations b 1 , h2 and h 3 are said to form a node. Making use of the definition eq. (1), we first choose the contour s near the point M 1 (fig. 6a) and then near the point M 2 . Since the total Burgers vector remains constant along the dislocation, we have b, = h 2 + h 3 • It is often convenient to use a new sign convention near the node. Let us take such a direction of a circuit S ds for each dislocation that it appears to be right-handed when looking outwards from the node (fig. 6b). Then the three Burgers vectors satisfy the symmetrical relation + h 2 + h 3 = O. More than three dislocations can meet at a node.
2.2. A dislocation cannot end within the crystal It is evident that the dislocation line cannot simply terminate within the crystal. It
must either reach the surface of the crystal at both ends or (as usually happens in actual cases) form a closed loop. In fact, if the dislocation line I ends at a certain point P in the medium (fig. 5), it is possible to imagine a surfaceZ, which spans the contour s, surrounds the end of the dislocation, and never crosses the line of singularities I. Then the line integral in the left part of eq. (1) could be transformed into the integral over the surface E*, duo j" . j" Ids = a dx, = j"-d s
2
aa au. 1
ckll1l
x,
Xl
x l1l
dE k ,
(2)
where Cikl is the antisymmetric unit tensor. The surface integral in eq. (2) is identically equal to zero, because the tensor Cikl is antisymmetrical. Since this result is in contradiction with the condition of eq. (1), the dislocation line cannot break suddenly withinthe crystal. It follows from this property of the dislocation that the Burgers vector is necessarily constant along the dislocation line. However, this statement should be made more accurate for the case when the dislocation line divides into several separate branches.
* The transformation of eq. (2) is made dl7l8 lk/
%x/, where
8 lkl
is
+ 1 if (i,
according to Stokes' theorem by replacing dx, by the operator k, I) is an even permutation of (123) and zero otherwise.
Fig.6
Three dislocations hI> bz and h 3 form a node. (a) A sign convention common for all dislocations. (b) A new sign convention near the node.
A stable array of dislocations in the crystal normally contains many nodes, so that the dislocations form a network. There are two- and three-dimensional dislocation networks. If only triple nodes are possible, the corresponding network must be topologically two-dimensional. But such a network can be distorted to a great extent and can be woven through its own parts. 2.3. Dislocation loops and plastic strain The condition of eq. (1) signifies from the mathematical point of view that in the presence of a dislocation the displacement vector is not a single-valued function of the coordinates, but receives a certain increment in a passage round the dislocation line. Physically, ofcourse, there is no ambiguity: the increment b denotes an additional displacement of the lattice points through one translational period of the crystal and this does not affect the lattice itself. In particular, the stress tensor Pik» which character-
A. M. Koseuich
42
Ch.l
izes the elastic state of the crystal is a single-valued and continuous function of the coordinates. The strain tensor ei k , which is usually determined as (3)
is also a single-valued and continuous function of the coordinates". Instead of the ambiguous vector of elastic displacement in the crystal with a single dislocation, we may always determine a single-valued vector u assuming that the function u(r) undergoes a uniform discontinuity b on some surface. For illustration, we consider a simple mechanism of the formation of an edge dislocation in the crystal. Assume that some shear occurs in the crystal and the corresponding displacement direction is shown in fig. 7 by arrows. As a result of this shear a part of the crystal
Crystal dislocations and the theory of elasticity
§2.3
Thus, we see that the function u(r) really has the fixed discontinuity b on some surface I~, spa~ni~g the dislocation loop. The surface Is may be chosen arbitrarily and the discontinuity of the vector u is found from «: - u" = h,
(4)
where u+ and u" are the values of u(r) on the upper and lower sides of the surface I respectively. The direction "upwards" (a positive one) is determined by the direction of the normal 11 to the surface Is (the relation ofthis direction with that ofthe vector tis shown in fig. 2).
I I
-
. Slip T
T
43
~
direction
'- - - -
A
A Fig. 8 A picture of the glide associated with the formation of the dislocation line AB. The crystal has slipped over the shaded portion of the glide plane.
Fig. 7 A model of the formation of an edge dislocation in the crystal.
above the plane TT' starts to slip over the lower part of the crystal and the upper part of the left side of the crystal (above T) is translated through a distance b to the right. If the glide has started at T and not yet reached the other end T' of the slip plane, then there is a region AA' on the slip plane over which n atom rows above the plane are associated with n - 1 atom rows below. This region contains the edge dislocation. All the atoms above the part TA of the slip plane are shifted by b relative to the atoms below this plane. Denote the quantity of Ux by u; and u; for the upper and lower sides of the slip plane, respectively. Then, U x = b in the region TA and u; = 0 in the region A'T'. Therefore the edge dislocation is the boundary of that part of the slip plane where the displacement vector undergoes a discontinuity. Its Burgers vector determines the magnitude of this discontinuity and the glide direction. Figure 8 shows a three-dimensional picture of the glide which is associated with the formation of the dislocation line AB. The right part of the crystal in fig. 8 corresponds to fig. 7. The displacement vector undergoes a discontinuity b on the shaded part of the slip plane.
o; -
u; -
* It should be noted that the representation of the ei k as the derivatives eq. (3) is not meaningful on the dislocation line, which is a line of singularities.
This fact allows one to give another formal definition of the dislocation, i.e. to describe it as a line I, which is spanned by the surface Is over which the vector u has a specified discontinuity given by eq. (4). In some cases this definition appears to be preferable to the first one of eq. (1). . I~ the strain tensor for the crystal with a dislocation is calculated by using eq. (3), r.e. III the presence of the discontinuity of eq. (4) along the surface I , it has a deltafunction singularity on this surface. The corresponding singular part of the tensor eik is given by e~~)
= i(nib k + nkbJ £5((),
(5)
where ( is a coordinate taken from the surface Is along the normal 11. Since there is no actual physical singularity in the space around the dislocation, the stress tensor Ps« must be a continuous function everywhere. According to Hooke's law, the tensor Pik is related with the elastic strain tensor e~ by: Pik
=
C ik1m erm'
(6)
Both tensorsrij and e~ should be continuous on the surface L s ' However, as mentioned above, the strain tensor has a singularity on this surface, which should not appear in the Hooke's law expression of eq. (6). Therefore, the total strain tensor eT in the crystal with a dislocation has the form of a sum ik (7)
44
A. M. Koseoich
Ch.l
The second item in the right-hand part of eq. (7), namely efk' is not directly related with elastic stresses. Such a strain is called plastic. Thus, the crystal with a dislocation may be considered to have a plastic strain efk = e~~), which is determined by the relation of eq. (5). It is of interest to note that the discontinuity surface 1:s may be chosen arbitrarily in a crystal with a single stationary dislocation. This signifies that the plastic strain is not a function of the state of the body, but depends on the method of formation of the dislocation or on the dislocation motion.
Crystal dislocations and the theory ofelasticity
§2.4
45
plane, and points «, {3, y, etc. determine atom positions in the succeeding atom layer. Then, vectors AB, AC, BC or IXP, lXy, py, i.e. joins of the nearest letters from the same alphabet, give the possible Burgers vectors of complete dislocations. The vectors yB and others which join the nearest letters from different alphabets give Burgers vectors of partial dislocations which are called Shockley partials. AB
2.4. Partial dislocations and stacking faults Dislocations are linear defects; there also exist in the crystal defects in which the regular structure is interrupted throughout a region near a given surface. In order to understand the physical nature of such defects, we consider a crystal to be made by stacking atomic layers. A displacement of one layer over its neighbour by a lattice vector lying in its plane does not alter the structure of the crystal. However, a displacement which is not a lattice vector produces a stacking fault. Such displacements, when combined with a small displacement normal to the stacking plane, can produce a configuration in stable mechanical equilibrium. Then the stacking faults become . stable. A well-known example of such defects is a narrow twinning band in the crystal.
x
Aa
IT
aB
Fig. lOAn extended dislocation. The complete dislocation AB splits into two partials AIX and IXB at the point X.
Now, consider a dislocation node, where one complete and two Shockley partial dislocations meet (fig. 10). All three dislocation lines and their Burgers vectors lie in the same crystal plane which is the plane of the figure. The complete dislocation AB splits into two partials AIX and IXB in a point x according to the scheme AB = AIX Fig. 9 Possible Burgers vectors in the close-packed structure. The closed circles represent atoms on a close-packed plane, and the open circles represent atoms on the succeeding close-packed plane.
The stacking fault can be macroscopically described as a surface of discontinuity over which the displacement vector u is discontinuous, but the stresses Pu. are continuous. If the discontinuity 0 is the same everywhere on the surface the resulting strain is just the same as that due to a dislocation along the edge of the surface. The only difference is that the vector 0 does not equal a lattice vector. The dislocation forming the boundary of the stacking fault is called a partial dislocation". The position of the surface 1:s discussed above in sect. 2.3, is no longer arbitrary, it must coincide with the actual stacking fault. Such a surface of discontinuity involves a certain additional energy which may be described by means of an appropriate surface-tension coefficient. The simplest types of partial dislocations exist in crystals with close-packed structures. Let points A, B, C, etc. in fig. 9 determine atom positions on a close-packed * A common dislocation where the vector b is a lattice vector is called a perfect dislocation or a complete dislocation.
+ IXB.
(8)
The region I of the slip plane where shear by a lattice vector occurs, and the region II where there is no displacement, are demarcated by the complete dislocation. The passage of the first partial dislocation along the slip plane causes a stacking fault which is healed by the passage of the second partial. Thus, the stacking fault in the region III is bounded by two partial dislocations. The configuration which arises of two partial dislocations separated by the stacking fault is called an extended dislocation. When two dislocations with Burgers vectors 0 1 and O2 combine to form a single dislocation with the Burgers vector 0 3 = 0 1 + O2 we have a dislocation reaction. The relation of eq. (8) illustrates a reaction of dissociation. A Shockley partial has its Burgers vector lying in the stacking fault plane. This dislocation is schematically drawn in fig. 11. A twinning dislocation is of the same type. However, there exist partial dislocations which have their Burgers vectors not in the stacking fault plane. Examples of such dislocations are shown in fig. I2a and b. The Burgers vectors are here perpendicular to the stacking fault plane. Such dislocations in crystals with a close-packed structure are called Frank dislocations.
46
A. M. Koseoich
Ch.l
Crystal dislocations and the theory of elasticity
§3.1
47
Thus, the problem of finding the many-valued function u(r) is equivalent to that of finding a single-valued but discontinuous function in the presence of the body forces given by eqs. (5) and (9). We can now use the formula
u,(r)
f
~ ~ik(r
r)f'f'(r) dr.
We substitute eq. (9) and integrate by parts; the integration with the delta function is then trivial, giving: uJr) = -Cjklmbm
Fig. 11 Shockley mobile partial dislocation.
rJE,.nl -00x ~i/r -
r') dE'.
(10)
k
Fig. 12 Examples of a sessile partial dislocation. (a) The positive Frank partial. (b) The negative Frank partial.
This solves the problem. In principle, eq. (10) allows us to obtain elastic displacements in the crystal for an arbitrary form of the dislocation loop. It should be noted, however, that the general formula of eq. (10) is very complicated and the calculation of the displacement field in an anisotropic medium even for simple dislocation line configurations is rather cumbersome and tiresome. In the case of a straight dislocation, the direct solution of the equilibrium equation under the condition of eq. (1) appears to be a more simple procedure. Equation (10) has a simple and evident form for the isotropic case. The elastic modulus tensor Cik1min the isotropic medium reduces to two Lame's elastic constants }c and fl (fl is the modulus of rigidity): Cik1m = A 6ik6lm + fl(6 a 6km + 6im 6kl ),
(11)
and the Green's tensor is
3. Elastic deformations in the presence ofa single dislocation
1 (26 ik }o + 02r ) ~ik(r) = 8nfl -r- - A + 2fl oX oX . i k
3.1. The displacement field around a dislocation loop The displacement field u(r) around a single dislocation can be expressed in a general form if we know the Green's tensor ~ik(r) of equilibrium equations for the anisotropic medium considered, i.e. the function which determines the displacement component U i produced in an infinite medium by a unit force applied at the origin along the xk-axis. This can be easily found by using the following formal device. Instead of seeking many-valued solutions of the equilibrium equations we shall regard u(r) as a single-valued function which undergoes a discontinuity given by eq. (4) on the surface spanning the dislocation loop I. We bear in mind that the strain tensor of eq. (3) has the singularity given by eq. (5) on the surface Es ' The strain tensor e~~) is formally related to a stress tensor p~~) = ciklmel~, which also has a singularity on the surface Es ' In order to eliminate this we must introduce fictitious body forces distributed over the surface E s with a certain density j(8). The equilibrium equations in the presence of the body forces are OPik/OXk + j~S) = O. Hence, it is clear that we must put oel~
-Cik1m ox
k'
(9)
(12)
After substituting eqs. (11) and (12) in eq. (10) and using Stokes' theorem, eq. (10) reduces to the following expression for the displacement of u in the point x: Q
u(x) = b4n Here
Q
+
1 4n
fb- rx -dl + grad lj/.
(13)
is the solid angle which the dislocation loop subtends at x (fig. 13) and
lj/ =
), + 4n(A
fl
+ 2fl)
f
dl b x (x - l) -, r
where dl is an element of the dislocation line, r = Ix - II and both integrals are taken . once round the dislocation line. The representation of eq. (13) for the displacement vector u in terms of the solid angle Q and contour integrals along the line I was derived by Burgers [2J. The line integrals are necessarily single-valued functions of x. But the vector u must be manyvalued for circuits linking the singular line, and all its multiplicity is due to the solid angle Q which is a multiple-valued quantity with the residue 4n.
A. M. Kosevich
48
Ch.l
Leibfried [3J has derived a similar formula for the dislocation in an anisotropic medium. The corresponding solution has the form: Q
u
= b- + u* 4n
(14)
'
where u* may be written as a certain line integral along the dislocation loop. However, the explicit expression for u* in an anisotropic medium can be obtained only for special cases. The problem can be solved when the Green's tensor C§ik is obtainable.
§3.1
Crystal dislocations and the theory of elasticity
49
The axial vector S has components equal to the areas bounded by the projections of the loop I on planes perpendicular to the corresponding coordinate axes. The tensor dik may be called the dislocation moment tensor. The components of the tensor C§ik decrease in inverse proportion to the distance 2 C§ik ex l/r, hence, it follows from eq. (15) that U i ex l/r . The corresponding strain field decreases with the distance according to the law of Pn. ex l/r 3. In an isotropic medium we can make use of eqs. (11) and (12); eq. (15) is then reduced to (17) where a = p/(A
x Fig. 13 The solid angle
Q
subtended at X by the surface L s ' which is bounded by the dislocation loop I.
The tensor C§ik for an anisotropic medium was derived by Lifshitz and Rozentsveig [4J. However, the derived expression appeared, in general, very complicated and the tensor C§ik was obtained in an explicit form [4J only for a hexagonal medium. The problem of obtaining the displacement field for an arbitrary dislocation line in a cubic medium was solved by Burgers [5]. If the departure from isotropy is small, the stress field of an arbitrary dislocation line in a medium of any symmetry can be found [3J by a perturbation method. Using eq. (13), the strain tensor around the dislocation in an isotropic medium can be calculated. The strain components are single-valued and thus eik can be represented by line integrals. Peach and Koehler [6J gave the corresponding expressions. These line integrals can be calculated explicitly if the dislocation line is a plane curve of the second degree (quadratic curve) [7]; in this case the integration leads to elliptic integrals. The deformation of eq. (10) has its simplest form far from the closed dislocation loop. If we imagine the loop to be situated near the origin, then at distances r, large compared with the linear dimensions of the loop, we have (15) where (16)
+ 2p).
Note the fact that the displacement field, far from the dislocation, is determined only by the symmetric part of the tensor dik . If the dislocation loop lies entirely in the plane in which the vector b lies, it appears convenient to choose the x-axis along b, and the z-axis perpendicular to the loop plane: b, = b 6ix ' Si = S 6iz ' Then eq. (17) will give us the expressions obtained by Nabarro [8J: UX
=
4~~3[a + 3(1 -
a) ;:Jz,
3bS uy = - 45 (1 - a)xyz, nr
UZ
bS [ a + 3(1 - a) rZ2J x. = 4nr 3 2
3.2. The strain and stress field of the straight edge dislocation" If the dislocation is a straight one, it is easy to ascertain the way in which strains and elastic stresses vary with the distance from the dislocation in the general case. In cylindrical polar coordinates r, e, z (with the z-axis along the dislocation line) the deformation will depend only on z and e. The integral of eq. (1) must, in particular, be unchanged by an arbitrary change in the size of any contour in the x-y-plane which leaves the shape of the contour the same. It is clear that this is true only if all the components of the tensor oujoxk are inversely proportional to the distance: ou) oX k ex l/r. The strain tensor eik and, therefore, the stress tensor Pu. will be proportional to the same power l/r. It was mentioned above that in the case of the straight dislocation when we deal with a two-dimensional problem of elasticity theory, it is sometimes not convenient to use eq. (10). It often appears preferable to solve the equilibrium equation directly.
* See Addendum 3.
50
A. M. Kosevich
Ch.l
As an example of the calculation of elastic fields in the presence of a straight dislocation, we determine the deformation around the edge dislocation in an isotropic medium [2, 9J. The physical meaning of this and other problems relating to an isotropic medium is purely hypothetical, since actual dislocations by their nature occur only in crystals, i.e. in anisotropic media. Such problems have illustrative value, however. We take coordinates x, y, z, so that the z-axis is along the dislocation line* and the x-axis is along the Burgers vector: bx = b, by = b, = O. It is evident from the symmetry of the problem with ez z = 0 that the displacement vector u lies in the x-y-plane and is independent of z so that the problem is a two-dimensional one. Love [1 OJ derived the following representation for the general solution of the twodimensional problem of elasticity theory:
Crystal dislocations and the theory of elasticity
§3.2
51
By using eq. (22) we can find the strain tensor of the edge dislocation e ik . This tensor has the following non-zero polar components: b« sin e err = eee = - 2n -r-'
b
ere = 4n(1
v)
cos 8 r
(23)
The stress tensor has the polar components: Prr
sin 8
= Pee = - D -r- ,
cos e Pe = D--, r r
(24)
where D = J1bj2n(1 - v). The Cartesian components of the tensor P« are
OX
2J1u x = 4(1 - v)P - ox' (25)
2J1u y = 4(1 - v)P -
oX oy'
(18)
where v is Poisson's ratio, and X = x(x, y) is the stress function which is equal to X = 2yQ
+
R.
(19)
Here, P, Q and R are harmonic functions; P, Q are harmonic conjugates. We wish to have a multiple-valued displacement Ux but single-valued stresses. Thus X must be single-valued, but P must be multiple-valued. Select a suitable pair of functions P and Q by choosing them to be the real and imaginary parts of the following complex function r(Z):
, = x + iy,
f(O = -iA log"
(20)
Since we assumed ez z = 0, then pzz = v(Pxx
+ pyy) = v(Pzz' + Pee)·
The average hydrostatic pressure produced by the edge dislocation in an isotropic medium is 2 sin 8 = 3(1 + v)D - - . r
1
Po =
3Pkk
(26)
Equation (26) has a simple physical meaning. Over the z-x-plane (8 > 0), where there is an "extra" half-plane inserted in the crystal, the medium is compressed, and Po > O. Below the z-x-plane (8 < 0) we therefore expect Po < O.
where A is any real constant. If we take R = 2Ay we obtain P
= Ai),
Q = -Alogr,
X
= -2Ay(log r
-
1),
(21)
3.3. Deformations near a screw dislocation and the symmetry of a medium
where
The cyclic constant A is obtained from the condition of eq. (1) and is A = bu] 4n(1 - v). Then the displacements are given by: b [ -1 Y u=-tan x 2n x
+
1 xY], 2(1 - v) r 2 2
uy
= - 4n(1 b_ v) [ (1 - 2v) log
* In all the
r
x ] + -;:z
.
problems on straight dislocations we take the vector t in the negative z-direction.
(22)
We determine the strains and stresses of a screw dislocation in an isotropic medium. Let the z-axis again be along the dislocation line; then the Burgers vector is b; = b; = 0, b, = b. It is evident from symmetry that the displacement u is parallel to the z-axis and is independent of the coordinate z. Since the relation Pu. = 2J1e i k for i .# k holds in an isotropic medium, the equilibrium equation in the absence of the body forces can be written in the form oeikjoxk = 0 and can be reduced to a twodimensional harmonic equation for Uz : Ju = 0, J = o2jox 2 + o2joy2. (27) z
The solution of eq. (27) which satisfies eq. (1) has the form b 2n
b tan -1Y -. 2n x
u = -8 = z
(28)
52
A. M. Koseoich
Ch.l
The tensors eik and Pu.have only the following non-zero components in the cylindrical coordinates:
j1b
b ezo = 4nr
Pzo
=
P yz
=
~[y
2n
ub [ 2n
(x -
X
O)2
x (x - X
+ y2
div
(x
+
- (x
+
-
Xo
O)2
+ y2
x
X
y+ ]
+ X
y2 '
O)2
Xo
O)2
+ y2
(J
O =,
(ja
=
au
"
Aap ;;--.
ux p
The required solution of this equation which satisfies eq. (1) is [14J: aa =
b
Paz
= 2n
)'ap8pzyXy
1111/2 1 - 1 A
Aa,p'Xa,x p'
IX
'
= 1, 2,
i;
1 where 1),1 is the determinant of the tensor Aap and is the inverse of the tensor AafJ • In the last formulae we sum over repeated Greek indices from 1 to 2.
3.4. The elastic energy of the
l;ltr~i1lioll1t
dislocation and the problem of thermodynamical
The elastic energy density is equal to -iPikeik' Therefore, the elastic energy per unit length of the straight dislocation is given by the integral
] .
53
only two components of the tensor Pik are non-zero: Pxz and pyz. We define a twodimensional vector (J and a two-dimensional tensor Aik : (ja = Paz; Aap = Cazpz(IX, fJ = 1, 2). Then the equation of equilibrium takes the form
(29)
= 2nr
Thus, the deformation around the screw dislocation in the isotropic medium is a pure shear. Since the mathematical theory of screw dislocations is rather simple, we can readily obtain the expressions for the stresses of a screw dislocation which is parallel to the plane free surface of an isotropic medium. Let the y-z-plane be the surface of the body, and let the dislocation be parallel to the z-axis with coordinates x = X o ' y = O. The stress field which leaves the surface of the medium a free surface is described by the sum of the fields of the dislocation and its image in the y-z-planes each considered to lie in an infinite medium: Pxz
Crystal dislocations and the theory of elasticity
§3.3
(30)
The dependence of the stresses in eq. (30) on the distance X o is typical of any straight dislocation near the crystal surface and is very important in discussion of various boundary problems. The comparison of the displacement of the screw dislocation with that around the edge dislocation (sect. 3.2.), shows that in an isotropic medium any straight dislocation has the following property. The dislocation with the tangent vector t may be considered to consist of two independent dislocations, i.e. a pure screw dislocation, the displacements of which are parallel to t, and a pure edge dislocation, the displacements around which are perpendicular to t, It is of interest to know whether the displacement field of a straight dislocation in an anisotropic medium can be split into a pure screw component, in which all displacements are parallel to the dislocation line, and a pure edge component, in which all displacements are perpendicular to this line. It appears that this is possible if the dislocation line is a two-fold rotation axis of the medium or if the dislocation line is perpendicular to a symmetry plane of the crystal. The elastic field of the straight dislocation in an infinite anisotropic medium was studied by many authors {Ll ] by using different methods. These methods were put into a systematic form [12J and were later extended for the case of a straight dislocation which is parallel to a plane interface between two anisotropic media [13]. The reader may find a detailed discussion of the corresponding problems in subsequent chapters. We confine ourselves to a simple example where the stresses are determined in an anisotropic medium around the screw dislocation which is perpendicular to a plane of symmetry of the crystal. We take the coordinates x, y, z in the usual manner and write Uz = u(x, y). Since the x-y-plane is a plane of symmetry, all components of the tensor Ciklm which contain the suffix z an odd number of times are zero. Thus,
U =
i f P;keik dx dy.
(31)
For a screw dislocation the integral of eq. (31) is: 2
Us
1 ="2
f 2pzoezo2nr dr
=
j1b fdr -, 4n r
(32)
where the integration over r should be made, in general, within the limits (0, 00). But the integral of eq. (32) diverges logarithmically at both limits. The divergence at the lower limit does not occur in real crystals because Hooke's law breaks down in the core of the dislocation. According to the existing atomic dislocation models, the lower limit r 0 should be of the order of the magnitude of the displacement that gives rise to the dislocation, that is ro ....., b. This estimate is natural from the point of view of elasticity theory, since it places a lower limit on the distance over which the dislocation theory considered here is valid*. Substituting the lower limit r0 in the integral of eq. (32) we omit the energy of the dislocation core Uo We easily estimate the maximum value of the energy Uo- The relative atomic displacements in the dislocation core with the cross section r~ ....., b? are of the order of magnitude of unity. Therefore the energy per unit length of the dislocation core is estimated as (33) The upper limit is determined by a quantity R of the order of the dislocation length or of the crystal size. We have then:
j1b 2 R Us = 4 log-· n ro
* We may call r 0
the radius of the dislocation core.
(34)
A. M. Koseoich
54
Ch. I
It is seen by comparison of eqs. (34)and (33) that eq. (34) determines the essential part of the dislocation energy only under the condition 10g(R/ro) » 1. One sometimes says that eq, (34) determines the dislocation energy with logarithmic accuracy. For the edge dislocation the integral of cq. (31) is
UE
~t
f
(p"e"
+ Po.eoe +
2p,.e,.)r dr dO
fR dr f21'C .
ub?
4n 2 (1 - v)
ro
r
sm
0
2
ede =
2
f1b log 4n(1 - v)
R -. ro
(35)
It is natural that the energy per unit length of the edge dislocation has the same order of magnitude as that of the screw dislocation. It is easy to estimate the energy per unit length for any slightly curved dislocation line with a radius of curvature R c which satisfies the condition 10g(Rc/r o) » 1. It is evident that the energy per unit length of such a dislocation has the order of the magnitude (see sect. 6.4.) V
r'V
ub?
R
4n
t
-log~·
§4.l
Crystal dislocations and the theory of elasticity
The movement of dislocations and plastic deformation of the crystal Conservative motion (glide) We refer again to the definition of the dislocation as a line which is the edge of a surface of discontinuity (eq. (4)). We used this definition earlier in a certain formal procedure which allows us to solve a number of static problems of elasticity theory in the medium with dislocations. Parallel to that we suggested a possible process of the formation of the dislocation by relative translation of atomic layers on the positive and negative sides of the surface 1:s through a distance b. However, some difficulties of a physical character arise from this manner of formation of the dislocation. In fact, when the requirement of eq. (4) is formulated we assume that the crystal remains continuous over the surface E s ' and, in particular, that the atomic spacings remain unchanged (with an accuracy up to elastic deformations). It is easy to understand that the actual discontinuity given by eq. (4) leads to failure of the crystal continuity.
(36)
«
Since unit length of any dislocation element has the energy of eq. (36), then there is a line tension T D of the dislocation, which is weakly dependent on the point on the dislocation line and may be estimated as TD
r'V
ub?
R
4n
ro
55
-log~'
n
(37)
It should be noted, however, that the theoretical "large parameter" 10g(R/r0) » 1 104 - 106 , although it is is not, in fact, so large. Really, if we take the ratio R/r o often smaller, then we come to the conclusion that log (R/r o) does not differ considerably from 4n. Therefore, the energy per unit length of the dislocation may be taken in a crude approximation as*
Fig. 14 A cut along the surface J:s ' The right cut surface undergoes a rigid displacement b. The closed circles represent atoms and the open circles represent vacant lattice sites.
r'V
(38)
Then the energy of an atomic length of the dislocation is approximately equal to
w = Ub ~ ub", 10- 2 3 cm ' and u 1011 erg/em" then w 10- 12 erg '" Since we have usually b 3 1 eV. This is a very large energy for each atom in the core of the dislocation. It appears [15, 16J that the increase of entropy in the crystal with a macroscopic dislocation loop cannot compensate such a large elastic energy. It follows that the free energy of the crystal with the dislocation exceeds that of the perfect crystal. Therefore, any dislocation which has a macroscopic length is thermodynamically unstable. Hence, the dislocated crystal can be in a state of mechanical equilibrium, but this state is a thermodynamically metastable one. r'V
* A similar estimate is valid
r'V
also in an anisotropic medium.
Really, the diagram (fig. 14) of the atomic arrangement along the part of the surface (under the condition nb > 0) shows that a macroscopic number of vacant lattice sites (the open circles in fig. 14) is formed after the relative translation of the atomic layers. The supply of some "extra atomic material" is required to fill these vacant sites and to ensure continuity of the medium. The additional volume associated with this material is equal to
b17 = nb b1: = b bE s
(39)
r'V
for each element bE of the surface of discontinuity. Therefore, the formulation of the condition of eq. (4) implies that we scrape away any surplus material where the atomic layers overlap and fill in any remaining gaps with the material supplied. But there are no mechanisms of automatic extraction or supply of the macroscopic quantity of the material inside the bulk of the crystal. Thus, a purely mechanical way of forming the dislocation is impossible by real displacement of the atomic layers on two sides of an arbitrary surface without causing discontinuities of the physical quantities on this surface. However, it follows from eq. (39) that for any dislocation in the crystal it is possible to find a surface 1:g spanning the dislocation loop in each point of which nb = O. It
Ch.l
A. M. Koseoich
56
is clear that this is a cylindrical surface with generators parallel to the vector band with a directrix along the dislocation line I (fig. 15). The displacement through a distance b over this surface is a sideways shift and does not break the continuity of the crystal. Such a surface is called the dislocation glide surface and is the envelope ofa family of the glide planes of all dislocation elements. By glide plane of a dislocation element is meant the plane tangent to the corresponding line element, a plane specified by the vectors t and b. Possible systems of glide planes in an anisotropic medium are determined by the structure of the corresponding crystal lattice. b
(a)
( b)
Fig. 15 Glide surfaces J.: g of dislocation loops. (a) The dislocation loop I may be reduced to a point by slip. (b) The dislocation I can make only a prismatic glide over the cylindrical surface with axis parallel to the Burgers vector b.
Thus, the displacement through a distance b over 1:g is "harmless" for the crystal, since lattice periods remain unchanged near this surface. In reality, one can shift the parts of the lattice on either side of the surface 1:9 to a distance of the lattice period b and form a dislocation. The physical peculiarity of the glide plane lies also in the fact that the dislocation can move comparatively easily in this plane. The latter follows directly from the microscopic picture of the dislocation defect and is readily demonstrated by the diagram with an extra half plane sketched in fig. 1. Let the edge dislocation be formed by the shift by a distance b along the glide plane. The trace of the slip plane coincides with the crystallographic direction TT' in fig. 16. We consider configurations near
§4.l
Crystal dislocations and the theory of elasticity
57
the dislocation core both for the first case, when the extra crystal half-plane is in the position MM ' (the atoms are shown by filled circles) and for the second case when the atomic layer in the position NN' plays the role of an extra half-plane of the crystal atoms are shown by open circles). Though the transition from the first atomic configuration to the second one is connected with the displacement of the dislocation by a single atomic spacing to the right of the glide plane, the displacements of individual atoms which make this transition possible turn out to be small as compared with b. This means that such a collective atomic rearrangement, which ensures the motion of the dislocation, can occur under the effect of comparatively small forces. The comparison between these forces and macroscopic stresses shows that the corresponding shear stress as which is necessary to start the dislocation motion is less than the rigidity modulus fl by a factor of 102 - 104 • The small value of the parameter asfu is the decisive physical factor which allows the use ofthe common linear theory ofelasticity for the description of the mechanical processes accompanied by the motion of dislocations. Thus, the dislocation can move rather easily in its glide plane in a purely mechanical manner. Such a motion of the dislocation is usually called glide or conservative motion. Dislocation loops may be divided into dislocations of two types according to their position on the glide surface. The dislocation loop of the first type (fig; 15a) may be made infinitely small by slip, i.e. it may be reduced to a point on the surface 1:g • The slip of the dislocation of the second type on the surface 1:9 (fig. 15b) would never make the dislocation loop infinitely small. In this case the motion of the dislocation on the glide surface is called prismatic glide. The dislocation loop of the second type has the smallest dimensions when it lies in the plane perpendicular to the vector b. The plane dislocation loop with its Burgers vector perpendicular to its plane is called a prismatic dislocation. However, a situation may arise when the dislocation can move by a slip into another glide surface (or to another glide plane if we have a plane dislocation loop). We consider a part of the dislocation line which lies in its glide plane (fig. 17a) and assume that it has a screw-type segment. Note that the plane dislocation loop with a smooth line (without breaks) has always a screw-type segment. The screw dislocation (t II b)
Ullb)
bF
GI:~I plane (a)
Fig. 16 The edge dislocation with the Burgers vector b undergoes a slip displacement along the glide plane TT'. An initial position of the "extra" half-plane is MM', and atoms are represented by the closed circles. A final position of the "extra" half-plane is NN', and atoms are represented by the open circles.
Fig. 17 Cross slip. (a) Primary glide plane of the dislocation line having a screw segment with t II b. (b) The screw segment bl o moves out of the primary slip plane into the cross-slip plane which also contains the Burgers vector b. A new position of the dislocation segment is b/ l .
A. M. Kosevich
58
Ch.l
can slip in any plane which contains it. Any plane parallel to b may be considered as a glide plane of the screw segment of the dislocation loop. This plane need not coincide with the glide plane of the rest of the dislocation crossing it at a certain angle. The screw segment with a finite length Mo can now slip in a new glide plane taking the shape of the arc M1 (fig. 17b). Such a deviation of the screw-type segment of the dislocation from the old glide plane into a new one is called cross slip.
Crystal dislocations and the theory of elasticity
6N is the difference in the number of interstitial atoms and vacancies generated. point defects for which eq. (41) is written appear and disappear directly near dislocation core. Therefore, all the changes of the crystal volume may be con\,j~AJ."J.'''''''''''''' on the dislocation line in the macroscopic description of the dislocamotion. The displacement of the dislocation in the direction perpendicular the glide plane should be accompanied by a local increase of the volume, and the relative volume change is 6e2k = 6x(b x t) 6(~),
4.2. Non-conservative motion (climb) The real motion of the dislocation in the direction perpendicular to the glide plane is of quite a different physical nature. We consider the arbitrary small displacement 6x of the element of the dislocation loop dl and refer again to the definition of the dislocation based on the condition of eq. (4). Irrespective of the choice of the initial surface LS' the real displacement of the dislocation element leads to an increase of the area of this surface by 6L, where
59
(42)
6(~) is the two-dimensional delta-function and ~ is the two-dimensional vector taken from the dislocation axis in the plane perpendicular to the vector at the point considered. The rate of the displacement of the dislocation described here is limited by the diffusion processes which ensure the volume change of eq. (42). Such a motion of the dislocation is called climb or non-conservative motion.
6I = 6x x dl. Such a change of the surface area L s is accompanied by an inelastic local increase of the crystal volume which, according to eq. (39), is given by
6T = b 6I = b(6x x dl) = -(b x dl)6x.
(40)
Note that the vectors band dl specify the glide plane orientation. In the case where the displacement 6x has a component normal to the glide plane, the quantity defined by eq. (40) is non-zero. It has already been mentioned that the lack (for 6T > 0) or excess (for 6T < 0) of material cannot be compensated in a mechanical manner inside the bulk of the perfect crystal if the continuity of the medium is conserved. However, in the perfect crystal there is a certain slowly acting mechanism of the change of the density of the substance, and this mechanism does not require macroscopic disturbances of the continuity of the medium. We mean the processes of formation and diffusion migration of point defects such as interstitial atoms (which densify the substance) and vacancies (which rarefy it). Therefore, dislocation motion in the direction normal to the glide plane can occur without disturbing the continuity of the medium only under the condition that there may be a diffusion exchange of matter between the dislocation line and the bulk of the crystal. This diffusion process can compensate an inelastic increase of the volume on the dislocation axis by the same decrease of the volume in the bulk of the crystal. The latter change of the volume is associated with the creation or destruction ofthe corresponding number ofpoint defects in the immediate neighbourhood of the dislocation core. Since each atom in the crystal has the relevant volume of the elementary cell a", the quantity eq. (40) should be related with the number 16TI/a3 of the vacancies created or interstitial atoms destroyed. But, as the point defects of both types can appear or disappear, the change of their number is associated with the displacement of the dislocation line element by the formula
6N =
b x dl - - 3-
a
6x,
(41)
Fig. 18 An originally straight screw dislocation AB with the Burgers vector b transforms into a helical dislocation.
The climb of a screw dislocation presents a rather interesting problem in the theory of dislocation motion. The straight screw dislocation AB (fig. 18) cannot move by climb as any plane is its glide plane if the line AB lies in this plane. However, if the dislocation is fixed in the points A and B, then any displacement of the dislocation line results in its curving. In the process of the screw dislocation curving there arises an edge component of the Burgers vector and, hence, some dislocation segments can now climb. The joint climb and prismatic glide may lead to the formation of a socalled helical dislocation (fig. 18).
4.3. The relation of the dislocation motion with plastic deformation The elastic field ofthe dislocation changes with its movement. However, the movement of the dislocation causes not only a change in the elastic deformation but also a change in the shape of the crystal which does not involve stresses, i.e. plastic deformation.
60
A. M. Kosevich
Ch.1
Crystal dislocations and the theory of elasticity
This is clearly illustrated by fig. 19, where the passage of the edge dislocation from left to right causes the part of the crystal above the glide plane to be shifted to the right by one lattice period; since the lattice is then regular, the crystal remains unstressed. Thus, dislocation motion is one of the mechanisms of plastic deformation.
61
n
20 The dislocation line I with the tangent vector t undergoes a displacement bX and occupies the position 1'. The ~-axis lies in the plane of the vectors t and c5X.
eqs. (45) and (46) we obtain the change of plastic strain, given by Fig. 19 A model of the atomic layer positions before, during and after the passage of a dislocation across the crystal.
de~
= i(nib k + nkbJ 6(0 dec;
rl",.. nl<:>lhup
Remember that, unlike an elastic deformation which is uniquely defined by the. thermodynamic state of the body, a plastic deformation depends on the process which occurs. However, if the motion of the dislocation is known, the change of the plastic deformation is defined uniquely. In sect. 2 we gave the formal definition of the plastic strain e~k over the surface 1:s spanning the dislocation loop I. Now consider a certain point on the dislocation line I and rewrite eq. (5) for the vicinity of this point as
efk = i(nib k
+ nkbJ 6(oe(~
- ~o),
(43)
where e(x) =
{I,0, xx <> O.0;
(44)
The (-axis is along the normal vector n, and the ~-axis is along the unit vector t x n (fig. 20) and the coordinate ~ is taken from the dislocation line, whose position is ~o
=
~o)'
(47)
from the definition of eq. (44) which determines the function e(x) that its equals the delta function: e'(x) = -6(X), and thus
de = (de/d~o) d;o =
-e'(; -
~o) d~o
=
6(~ - ~o) d~o.
(48)
substituting eqs. (45), (46) and (48) in eq. (47) we have 6e~
= -i(biCklm + bkCilm)tm 6(,) 6X1 ,
(49)
6(~) is the same two-dimensional delta function as in eq. (42) and ~ is the two-drmensronat radius vector with the components ((, ~ - ~o)' see that, in fact, the change of plastic deformation is related uniquely with the parameters of the dislocation, i.e. the vectors band t. and with the value of the dislocation displacement 6x. The plastic deformation related with the motion of a single dislocation should be taken into account when one dislocation cuts the line of another. We consider two dislocations: one of them moves and the other is immobile. Let the line Ii be the original position of the first dislocation (fig. 21), 1: ~ be its glide surface and the line AB
~o(l)·
If the dislocation is immobile, the surface 1:s and hence the vector n may be taken arbitrarily. If a real displacement of the dislocation takes place, the change 61:s is determined uniquely and the corresponding normal n also becomes single-valued. Let 6x be the displacement of the dislocation which causes the dislocation line to take the position i' (fig. 20). The ;-axis lies now in the plane which is specified by t and 6x. We shall derive the formula for the change of e ~ with such a motion of the dislocation. We have 61:s = 6x x dl and
n = (6X x t)/16xl sin cP,
(45)
where cP is the angle between the vectors 6x and t (fig. 20). It is also evident that 6~o
= 16xl sin cP.
(46)
Fig. 21 The dislocation line AB with the Burgers vector b is cut by another dislocation which has the Burgers vector b1 perpendicular to the line AB and moves from a position 11 to a position I~. As a result a step AOO/C arises. This step can be resolved into a kink and a jog.
Ch. 1
A. M. Kosevich
62
be the original position of the second dislocation. As a result of the displacement by bx, the first dislocation cuts the line of the second dislocation in the point 0 and occupies the position 1'1 (fig. 21). Then points on the line of the second dislocation, separated by the glide plane of the first one, suffer a relative displacement of hi. This displacement appears as a step AOO'C in the line of the second dislocation". We can resolve the step h1 in the line of the second dislocation into three orthogonal components. The first one is parallel to the dislocation line. The second, in the glide plane of the second dislocation 'E9 and perpendicular to its line, is called a kink. The third component, normal to the glide plane of the second dislocation, is a jog (fig. 21).
Crystal dislocations and the theory of elasticity
63
dislocation is not related with the additional body forces within the crystal, = 0 and the first integral on the right of eq. (52) vanishes. In order to ""aI'...,....A'... ~.., the last two integrals we take for 'E a surface, made up of the upper and surfaces of the cut l:p separated by a gap h and connected by a tube 'Ep whose is the dislocation line I and whose radius is p. The scheme ofthis surface is sketched 22. 0PidoXi
°
5. Interaction of dislocations with an elastic field 5.1. The action of a stress field on a dislocation Let us consider a dislocation loop I in a field of external elastic stresses Pik created by given loads, and find the force on the loop in such a field. We calculate the work bR done by external loads in an infinitesimal displacement of the loop I. If this work is written as (50)
bR = tFbxdl,
where bx is the displacement of the dislocation line element, then Fwould determine the force on unit length of the dislocation line. Let the displacement of the dislocation cause a certain change of the displacement vector bu. Then the work of the external stresses done on the volume 1', which includes the dislocation loop, is given by
bR =
f»«
(51)
bu, dE ;00,
where 'E is the closed surface bounding the volume 1'. It is convenient here to use transformations of the integral of eq. (51) which are based on Gauss's theorem and therefore the displacement u around the dislocation should be considered as a single-valued function of the coordinates. In that case this vector will have a discontinuity on the surface 'Es spanning the dislocation loop. In order to eliminate the points of discontinuities of the vector u, we assume the loop I to be enveloped by a certain closed surface 'E° outside which the function u = u(r) is continuous (in the volume 1"). Then (jJ
s« ~ =
f [O(P'k bu,)/ox,] dr ' - f P;, bu, dE?
,.
J
(oPidoXk) bU i dr
,.
+ JPik s-; dr + JPik bukn? d'E°,
where the normal nO is directed outwards from the surface 'E 0.
* There
is a similar step of b in the line of the first dislocation.
Fig. 22 The dislocation line I is enveloped by a tube It which has radius p and joins the upper and lower surfaces of the cut Is.
If we neglect the local perturbation of the crystal which is caused by the dislocation along its axis and is insignificant in the estimation of the interaction of the dislocation with the elastic field, the remaining volume integral on the right of eq. (52) can be reduced to the integral over the total volume 1'. In fact, as the stresses and strains are continuous functions, the integral over the volume 1" turns into the integral over the total volume 1', when one surface of the cut 'Es approaches the other (h -+ 0) and the radius of the tube decreases (p -+ 0):
fPik s-; fPik dr ' --,
(53)
The surface integral in eq. (52) can be also simplified by a relevant limit transition. In particular it should be noted that the integral over the tubular surface 'Et becomes zero as p -+ 0, because the displacement field of the dislocation has the following feature lim pu(r) = O. p-tO
The values of the continuous quantities P« are the same on both surfaces of the cut, but the limit values of u differ by a given amount h. Therefore, instead of eq. (52) we have
bR = (52)
beik dr.
fP;, be;, dr + i JPlI be
kk
dr
+ b;b JP;k az,
(54)
where P;k is the stress deviator, e;k is the strain deviator (e;k = eik - :tbikel/) and the first two volume integrals replace the integral of eq. (53) according to the evident identity Pikeik
= (Pik - :tbikPll)e ik + :tPllekk = P;ke;k + :tPllekk·
A. M. Koseoich
64
Ch.l
Since the distribution of the stresses Pik is assumed independent of the position of the dislocation, we take the difference symbol £5 outside the last integral in eq. (54). If the element of the dislocation line dlis displaced by an amount £5x, then the area of the surface Ls changes, the elementary change being (55) The relation of eq. (55) should be used in the transformation of the last integral in eq. (54). In the case where the displacement £5x lies in the dislocation glide plane, eq. (55) characterizes the main changes in the crystal. However the situation is different in the presence of a displacement which is perpendicular to the glide plane. In this case, the additional requirements which follow from the continuity of the medium become rather essential. It is usually of interest to study those motions of the dislocation which take place without disturbing the continuity of the medium. Then, as shown in sect. 4.2., the displacement of the dislocation is accompanied by a local relative change of the volume which is given by eq. (42). Using this equation for the transformation of the second integral in eq. (54), we may write (56) Now, with eqs. (55) and (56) in mind, we can separate that part of the work £5R which is associated with inelastic deformation of the medium arising from the displacement of the dislocation. Write eq. (54) in the form
~R =
JPik
~e;k dr + ts;","I",(P"k - 4(j"kPll)b k ~X; dl.
ts;","I",P;kbk
~x; dl
FEy = _b'Plxx --
-~b(2.p 3 xx
- Pyy - Pzz ) .
65
(61)
It is of interest to determine the component of the force of eq. (59) lying in the
plane of the corresponding element of the dislocation. Let K be a vector normal the dislocation line in the glide plane. Then the required force component (f, say) = KF = Cikl KJkPlmbm or (62) where n = K X t is the normal to the glide plane. Since the vectors nand bare perpendicular, we see that the force f is determined by only one component Pik if two of the coordinate axes are taken along these vectors. If the dislocation forms a plane curve which lies entirely in its glide plane and the external elastic field is uniform, then the force f is the same for all the elements of the dislocation line irrespective of their positions on the glide plane. The last fact is used in the quantitative description of a Frank-Read source (see sect. 5.6.). Since the force of eq. (62) may cause motion of the dislocation in its glide plane its magnitude would determine the start of plastic deformation in the crystal. Equation is in agreement with the experimental result that slip begins when the shear stress niPikbk/b reaches a value which is characteristic of the crystal, but independent of the other components of the applied stress. The total force acting on the whole dislocation loop is
(57)
The first integral in eq. (57) is equal to the increase of the elastic field energy in the body. The linear integral over the dislocation loop, namely
~RD =
Crystal dislocations and the theory of elasticity
(63)
This is zero except in the case of a non-uniform field' when PI = constant the integral is f dx, = O. If the stress field varies only slightly ~ver the 'loop, we can 'write:
(58)
yields the work of the displacement of the dislocation. Comparison of eqs. (58) and (50) shows that the force on unit length of the dislocation is
This force can be expressed in terms of the dislocation moment dk1 defined by eq. (16):
(59) A formula of the kind eq. (59), where the stress deviator P;k is replaced by the stress tensor Pik was derived first by Peach and Koehler [6]. Weertman [17J pointed out the necessity of taking into account the inelastic change of volume in dislocation climb, i.e. the necessity of replacing Pu. by Pi« _. j £5 ikPll' As a trivial example of the use of eq. (59), we consider the forces on screw and edge dislocations. Let the z-axis be parallel to the dislocation line (tz = -1). In the case of the screw dislocation b, = b, and
F; =
(64)
If there is a uniform stress field near the dislocation loop and F T = 0 the dislocation
is acted upon by a couple with the moment
(60)
which also can be expressed in terms of the dislocation moment of the dislocation loop:
In the case of the edge dislocation the x-axis is taken along its Burgers vector ib ; = b).
(65)
F; = bpyz'
-bpxz'
A. M. Kosevich
66
Ch.l
We consider a plane circular prismatic dislocation which has the Burgers vector b perpendicular to the loop plane (along the z-axis). Such a dislocation is acted upon by the following moment of forces:
where S is the area of the loop.
Crystal dislocations and the theory of elasticity
§5.2
The force between two edge dislocations is not, generally speaking, radial and, besides, has essential angular dependence even in an isotropic medium. We assume the glide planes of the dislocations to be parallel. Let the z-x-plane be parallel to the glide planes and the z-axis be parallel to the dislocation lines. Then calculations may be made directly by using eqs. (61) and (25). If one dislocation is along the z-axis (fig. 23), it exerts on the other dislocation (passing through the point (r, 8) on the x-y-plane) a force whose component in the glide plane is
5.2. Forces between straight dislocations The Peach-Koehler formula (eq. (59)) allows us to describe the interaction between two dislocations in an obvious way. If Piker) represents the stresses produced by one dislocation near an element dl of the other dislocation having Burgers vector b, then the relationship (66) determines the force exerted by the first dislocation on the element dl of the second dislocation. However, eq. (66) is useful only in the description of the interaction between dislocation lines of simple shapes and orientations. For example, it is easy to analyze the interaction between two dislocation loops, if the distance between them considerably exceeds the dimensions of the loops, For that it is enough to substitute the stress tensor obtained by using eq. (15) in eqs. (64) or (65). But consideration of the force of interaction between two closely situated dislocations would require cumbersome calculations. Often it is more convenient to characterize the interaction between arbitrary dislocation loops by their interaction energy (see sect. 6.4.). Note that only the interaction of straight dislocations can easily be considered in the general case. We shall calculate the forces between two parallel straight dislocations in an isotropic medium. The interaction force between nonparallel dislocations was calculated by Kroupa [18]. We begin our considerations with the simpler case of two screw dislocations. The force per unit length acting on one dislocation in the stress field due to the other dislocation is determined from eq. (60), using eq. (29). It is a radial force of magnitude
= f.1 b I b 2 .
F r
67
b M F; = b 12
cos 8 cos 28
r
'
M
=
f.1/ 2n (1 - v).
(68)
It is this component that presents the greatest interest, since the displacement of the dislocation may occur as a mechanical motion only in the glide plane. The equation of this component ofthe force to zero corresponds to the equilibrium configuration ofthe two dislocations relative to slip. It follows from eq. (68) that F; is zero at 8 = 0 and () = It can be easily verified that the first position (fig. 24a) satisfies the requirement for the stable equilibrium of two dislocations with like signs (b i b2 > 0), while the second one (fig. 24b) does so for the two dislocations with unlike signs (b i b2 < 0). It is evident that the first configuration (fig. 24a) also remains in equilibrium if there are more than two dislocations. This is an explanation of the fact that straight edge dislocations which lie in parallel glide planes have a tendency to gather in one plane perpendicular to their glide planes. The array of parallel edge dislocations placed in the plane perpendicular to their Burgers vectors is called a dislocation wall. The tendency of dislocations to form isolated dislocation walls leads to polygonization. The equilibrium configuration in fig. 24b is the basis of the formation of double dislocation lines, the so-called dipoles. Two parallel straight edge dislocations which lie in the same glide plane (8 = 0) interact with the force
in.
F x
=
b lb 2M r
=
-
b lb 2M
IX I
-
x 21
,
(69)
where Xl and X 2 are the coordinates of the first and second dislocations, respectively. Equation (69) shows that dislocations of like sign (b I b2 > 0) which lie in the same glide plane repel, while those of unlike sign (b I b2 < 0) attract.
(67)
2nr
y
Screw dislocations oflike sign (b i b2 > 0) repel, while those of unlike sign (b i b2 < 0) attract. y
x
o
e o Fig. 23 The component F; of the force on a dislocation at (r, 0) produced by a dislocation at the origin.
(a)
Fig.24
(b)
The stable equilibrium positions of two dislocations. (a) The dislocations have like signs. (b) The dislocations have opposite signs.
A. M. Kosevich
68
Ch.l
A similar expression yields the interaction force between edge dislocations which lie in the same glide plane in an arbitrary anisotropic medium. However, the coefficient M, in this case, is expressed in terms of the elastic moduli of the medium in a more complicated way. Finally, we pay attention to the problem of forces between straight dislocations which form a node of the dislocation network. The theory of a regular dislocation network was given by Frank [19]. It was extended later by a number of workers, and was verified experimentally (see e.g. [20]). The accurate expressions for the forces between three dislocation lines, which converge in one point, were derived by Indenbom and Dubnova [21]. They demonstrated the invalidity of the linear tension approximation (see sect. 3.4.) in this problem. The equilibrium configurations of the dislocations in such a node were also obtained [21].
Crystal dislocations and the theory of elasticity
§5.3
from eqs. (61) and (25) by summing the contributions from all the dislocations in the wall: L X2 - (y - 11)2 d1] 2LbD x(x 2 - y2 + L 2) 2 2 F; = bDx -L [x + (y _ 11)2]2 -; = -w- (x _ y2 + L2)2 + 4X2y2' (70)
f
The force of eq. (70) tends to pull the dislocation into the plane of the wall or to drive it out according to whether (x 2 - y2 + L 2) is positive or negative. In fig. 25 the region of attraction is bounded by the hyperbola. Thus, the dislocation wall which nucleated in the crystal automatically provides the conditions for its further growth. It is of interest to calculate the shear stresses produced by an infinite dislocation wall over large distances. The element Pxy of the stress tensor produced by all dislocations of the wall* is given by a sum
5.3. Pelygonizatlon and a dislocation model of grain boundaries
00
Pxix,y)
It is not difficult to understand that a uniform distribution of edge dislocations with parallel Burgers vectors is unstable relative to the formation of dislocation walls. We consider a dislocation system containing a large number of parallel edge dislocations which can move in their glide planes. It follows from eq. (68) that the force on an isolated dislocation exerted by other uniformly distributed dislocations is zero. Therefore, a nonzero force may occur only owing to the local disturbance ofthe symmetrical configuration of neighbours near the dislocation under consideration. Then, two or three dislocations arrange as in fig. 24a, giving an origin for the formation of the dislocation wall. Suppose that the process of building up the wall has continued until a wall. of length 2L has been formed (fig. 25). The dislocations are assumed, for simplicity, to be at a uniform spacing ofw. We find the force in the glide direction on a dislocation at a point (x, y) at a distance from the wall large compared with w. This force is obtained
\ \
\
\ \ \
\ \
x
'\
01. -, I
\
\
/
\
/ I
T;
-Lw
/
/ /
/
-L
\
\ \
\
69
= Dx
n=~
00
x2 [x 2
_
+
(y _ nw)2 (y - nw)2]2'
(71)
For x » w the sum reduces to Pxy
= 411?D(x/w 2) e-2nxjw cos (2ny/w).
(72)
Thus the stresses decrease exponentially away from the wall. Therefore, unlike the field of an isolated dislocation, the elastic field of the wall is concentrated in a thin layer, the thickness of which is of the order of the spacing between the dislocations. It is easy to understand the geometrical meaning of the crystal deformation produced by such an array of dislocations. The presence of the dislocation wall leads to the misorientation of the two parts of the crystal separated by the plane of the wall (fig. 26). If w is the distance between the dislocations then the misorientation angle of the two parts of the crystal is l/J = b /w. It is necessary in the macroscopic theory that w » band l/J « 1. The dislocation wall may be considered as a model of the boundary between two blocks or subgrains with a slight misorientation. If the boundary consists of edge dislocations, then the axis about which the neighbouring subgrains have been tilted lies in their interface. Such a boundary is known as a tilt boundary in contrast to the twist boundary in which the two grains are rotated about the normal to their interface. The twist boundary contains a certain grid of screw dislocations. The arrangement of the dislocations in the grain boundaries formed in the manner described is called polygonization. The formation of regular boundaries involves both dislocation glide into the neighbourhood of the boundary, often by large distances, and the climb of dislocations normal to the glide plane by short distances. The latter is required to form a reasonably uniform distribution of dislocations in the . boundary**. It was mentioned that elastic stresses decrease exponentially with increasing distance from the dislocation wall. This means that the single crystal remains unstrained on
\
Fig.25 A dislocation wall and its action on a dislocation at a point (x, y). The force in the glide direction tends to attract the dislocation into the plane of the wall above and below the two branches of the hyperbola.
* The detailed analysis of all the elements of the stress tensor due to the dislocation wall is given in works by Li [22]. ** At low temperatures when glide but not climb is possible, only short boundary segments are formed.
Ch. 1
A. M. Kosevich
70
71
Crystal dislocations and the theory of elasticity
§5.4
~Em 1r----~~--~
o Fig.27
Fig. 26 A dislocation model of a tilt boundary: two grains inclined at a small angle lj/.
both sides of the boundary, and its lattice is regular. All the elastic energy associated with the junction of two slightly misoriented parts of a single crystal is concentrated in a narrow layer near the grain boundary. The energy of the subgrain boundary may be easily estimated as a function of the misorientation angle t/J . We use the fact that the elastic field of the wall decreases exponentially at distances x » w and coincides with the field of an isolated dislocation at x 2 + (y - nw)2 « w 2. Then. for a rough estimate we may regard the elastic energy of the wall as the sum of contributions from the isolated dislocation, supposing each single dislocation to be situated along the axis of a cylinder of radius iw. The energy per unit length of this dislocation is equal [see eq. (35)J to
tlb2 W ----log-, 4n(l - v) 2ro
We suppose for definiteness that the z-axis is parallel to the dislocations, the x-z-plane is the glide plane and the Burgers vectors of the dislocations are in the x-direction. Then the force in the glide plane per unit length of a dislocation is bpxy, where Pxy is the stress at the position of the dislocation. We consider a set of n edge dislocations in which the leading dislocation is held up at the point Xl = 0 (fig. 28), and remaining dislocations are in equilibrium at points X 2 . . . X n such that
l.
btl t/Jo E(t/J) = 4n(1 _ v) t/J log T'
(73)
where t/Jo = ib/r o· The energy of the subgrain boundary of eq. (73) as a function of t/J is illustrated in fig. 27. ThefunctionE(t/J)hasacuspatt/J = oand a maximum Em att/J = = t/Jo/e. The quantitative agreement of the relation of eq. (73) with experiments is discussed in refs. [23, 24J.
v;
5.4. Planar dislocation pile-ups Let us consider a large number of similar straight dislocations lying parallel in the same glide plane, and derive an equation to determine their equilibrium distribution.
1
ffi
.Lx 8
Fig. 28
where the radius of the dislocation core ro is independent of t/J provided that t/J is reasonably small. The distance between neighbouring dislocations is w = b/t/J and, hence, the energy per unit area of the grain boundary is
The grain boundary energy E(lj/) as a function ofljJNm'
Dislocation pile-up. The leading dislocation is held up at the point x the origin and crosses the glide line at a point x.
=
O. A circuit T surrounds
The problem is to find the law of the distribution of dislocations along the x-axis under a given external stress. This distribution may be obtained from the condition of equilibrium for each dislocation subjected to an external force and the force of interaction with other dislocations in the set. Taking into account the interaction force of eq. (69) and the force on the dislocation exerted by the external stresses, we can write the condition of equilibrium" as
±
=
D
i= 1,i*k Xi -
Xk
P~/Xk)'
k
= 1,2, ... n,
(74)
where P~y(x) is the external shear stress, and D = tlb/2n(1 - v). The problem of the dislocation distribution is thus reduced to the resolution of the nonlinear system of algebraic equations determining the X k. * See Addendum 5 and 8.
A. M. Koseoich
72
Ch.l
A special method for the solution of this system was suggested by Eshelby, Frank and Nabarro [25]. It appears that if we introduce a polynomial of degree n which has X k as its zeros, this polynomial satisfies a certain differential equation of the second order with variable coefficients. Thus, e.g. if the position of the first dislocation is fixed* and the external stress is uniform, the coordinates of the remaining dislocations in the set will be the zeros ofthe derivative of the n-degree Laguerre polynomial. Using the asymptotic expression at large n for the zeros of the Laguerre polynomial, Stroh [26] later calculated the stresses due to such a dislocation pile-up. We may determine the solution of the problem of the distribution of n dislocations more directly by proceeding from the assumption that the number of the dislocations in the pile-up is rather large and the dislocations are continuously distributed over their glide plane. Let .@(x) be the line density of dislocations on a segment (L 1, L 2) ofthe x-axis: b'@(x) dx is the sum ofthe Burgers vectors of dislocations passing through points in the interval dx. Then the total stress at a point x on the x-axis due to all the dislocations is given by the integral**
__I
L
PXy(x) -
D
2
L1
.@(x') dx'. -,-X - x
(75)
For points in the segment (L 1 , L 2 ) this integral must be taken as the Cauchy principal value in order to exclude the physically meaningless action of a dislocation on itself. The condition of equilibrium of a dislocation at the point x has the form t
p
I
L2
L1
.@(x') dx'
r; (x)
= _x_y_ == (V (x ),
x' - x
D
(76)
where P denotes, as usual, the principal value. This is an integral equation to determine the equilibrium distribution .@(x). It is a singular equation with a Cauchy kernel. Such an approach to the problems of dislocation pile-ups was first suggested by Leibfried [27]. The function .@(x) satisfies the following explicit normalization requirement ~(x)
dx = n.
(77)
L1
The theory of singular integral equations with a Cauchy kernel was discussed in detail by Muskhelishvili [28] and we confine ourselves only to some certain results of this theory [14]. Let us consider the case where there are no external stresses ((V(x) = 0) and the dislocations are constrained by some obstacles (lattice defects) at the ends of the segment (L 1, L 2). For (V(x) = 0 eq. (76) has a nontrivial solution ~(x)
= n/n[(x - L 1)(L2 - X)]1/2,
* This may be, for ** See Addendum
t
example, a so-called sessile dislocation. 5.4. See Addendum 5 and 8.
73
which describes a pile-up of n dislocations. We see that the dislocations pile up towards the obstacles at the ends of the segment with a density inversely proportional to the square root of the distance from the obstacles. The stress outside the segment (L 1 , L 2 ) increases in the same manner as the ends of the segment are approached, e.g. for x > L 2 Pxy ~ nD/[(x - L 1)(L2 - L 1)r /2. In other words, the concentration of dislocations at the boundary leads to a stress concentration beyond the boundary. If the dislocations are constrained only by external stresses, then the function ~(x) is given by the formula:
1 ~(x) = - n2 [(L 2 - x)(x - L 1)]1/2 P
I
L
L1
2
[(L
2
(V(x') - x')(x' - L
1)]1/2
dx' x' - x
(78)
The solution of eq. (78) has the following feature: '@(L1) = ~(L2) = O. Equation (76) may have a solution of this kind only under the condition [28] (V(x) dx = 0 x)(x - L 1) ] 1/2 .
(79)
Finally, we describe the dislocation distribution on a segment with an obstacle at one fixed end of the segment (say, at a point L 1 ) . For simplicity, we take L 1 = 0 and L 2 = L. Then the solution will be ~(x)
X)1/2 = - -12 (L --P rr
x
If the stress field is uniform eq. (80) gives
P0 ~(x) = _
L2
I
Crystal dislocations and the theory of elasticity
§5.4
rrD
f.L ( 0
(p~ix)
(L X)1/2 . _-_ x
x'
L - x'
)1/2 (V(x') dx' . x' - x
(80)
= -Po), then the calculation of the integral of
(81)
This function is sketched in fig. 29. The condition of eq. (77) determines the length of the dislocation pile-up L = 2nD/po. Beyond the obstacle there is a concentration of stresses near it given by Pxy ~ - Po(L/lxj)1/2.
(82)
Relations of the type of eqs. (81) and (82) were used by Stroh [26] to calculate the stress concentration in the neighbourhood of the head of the dislocation pile-up. In conclusion we determine the difference in displacements in the glide direction on the opposite sides of the plane surface along which the continuously distributed dislocations are situated. Consider a circuit T surrounding the origin of coordinates and crossing the glide plane at a point x (fig. 28). Take u(x) = »; u; where u:
* See Addendum 5.4, 2.
74
A. M. Kosevich
Ch. I
D9;J(x)
TIpo
o Fig.29
The dislocation density
~(x)
as a function of xfI: The concentration of the dislocations at the head of the pile-up is shown.
and u; are the crystal displacements at a point x above the glide plane and under it, respectively. Then, with the definition of a dislocation eq. (1) we have u(x)
=
lJr ~u~ dx] =
-B,
Crystal dislocations and the theory of elasticity
§5.5
75
the resistance of the crystal to the displacement of the dislocation. The magnitude of the frictional force depends on the type of the dislocation and to a considerable degree, on the model of the dislocation core. To study the role ofthe discrete structure of the crystal lattice, Peierls [29J suggested a simple model which now has his name. The Peierls model for the edge dislocation is, in essence, the following. Let the atoms of the undistorted crystal form a simple tetragonal lattice with a period b along the x-axis and with a period a along the y-axis (fig. 30). We consider two parts of such a crystal, one of which (A) has an extra atomic plane as compared with the second part (B). Put these two parts together and form a single crystal by bringing together upper and lower parts at the lattice parameter a and by matching left and right edges of the two parts. To achieve the latter we may fix the right edges of the parts A and B and then cause a relative shift of their left edges along the x-axis by the distance b. When the two parts of the crystal combine, the atoms form a single lattice which has an edge dislocation (fig. 30b).
(83)
UX k
where B is the total Burgers vector of the dislocations embraced by the circuit T': B = b
I:
A
!'0(x') dx '.
(84)
\
L
It follows from eqs. (83) and (84) that u(x) =
-i
I:
=
!'0(x') dx.
I du(x) ----. b dx
-x 8
(a)
Thus, the dislocation density is determined by the derivative of the relative displacement of the crystal on the opposite sides of the slip plane: ~(x)
r-
(b)
Fig. 30 The formation of a single crystal from two separated parts A and B. (a) The part A has an extra atomic plane as compared with the part B. (b) The two parts A and B are joined and make a united single crystal with an edge dislocation.
(85)
5.5. The Peierls-Nabarro force In preceding sections we discussed the force on any dislocation in a continuous medium. This force arises from the elastic stress field. However, a dislocation in a crystal is also acted upon by forces which cannot be described by elasticity theory. We call these forces, which are "extraneous" to elasticity theory, forces of inelastic origin, and discuss briefly only two types of such forces. Inelastic forces of the first type act on partial dislocations. A single partial dislocation is a contour of some planar stacking fault in the crystal, and it is acted upon by the surface tension connected with the surface energy of the defect. This force is always directed in the plane of the defect normal to the dislocation line at a given point. The contribution of the surface tension is very important in writing the equation ofequilibrium of a thin twin in the crystal (see sect. 5.7). Inelastic forces of the second type are due to the discrete nature of the crystal and the atomic character of the structure of the dislocation core. These forces describe
We denote the relative atomic displacement of the two sides of the glide plane in the direction of the x-axis by u(x). The crystal lattice should be perfect at infinitely large distances from the dislocation, and this is possible if the condition
u(oo) = 0,
u( - (0)
= b
(86)
ib
is satisfied. We take u(X) = and suppose the center of the dislocation to be in a point x = X. To determine accurately the function u(x) we assume that the upper and lower parts of the crystal may be considered as two elastic continuous half spaces; however, the tangential stress in the interface is supposed to be a periodic function of the local relative displacement u(x) with period b. Peierls took the simplest periodic function, namely, a sinusoidal function: P
=
III sin (2nu(x)/b) ,
(87)
where the coefficient /11 has the order of magnitude of the elastic modulus of a crystal. We shall show below the way in which it is determined.
Ch.l
A. M. Kosevich
76
The local stresses of eq. (87) must coincide with the macroscopic stresses caused by the nonuniform distribution of the displacement u(x) along the whole glide plane. We have already seen that a nonuniform relative displacement u(x) on the opposite sides of any plane is equivalent.to a certain distribution of edge dislocations along this plane with the density of eq. (85). Therefore, it causes the shear stresses of eq. (75). Substituting eq. (85) in eq. (75) and comparing the result with eq. (87) for plJj(x), we obtain a singular integral equation for u(x): P
foo _ 00
du(x') dx' dx x - x'
= _ fll b sin (2TC U(X)). D
b
D
x
foo dU(~') dx' -00
dx
= ~.
Let us now calculate the derivative with eq. (90). Then we find
oUx/oy at y =
°by using eq.
(90) (22) and compare it
_ 00
du(x') ~ = _ 2(1 - vl~ sin [2TC U(X)]. dx' x - x' 3 - 2v a .b
(91)
The solution corresponding to the conditions of eq. (86) is u(x) =
~b
(1 _
2 tan-l x -: X),
tt
(92)
J,
where
3 - 2v
), = 4(1 _ v) a.
Jo
sin (2TCU/b) du (94)
Now, take into account the discrete structure of the glide plane and denote the coordinates of atomic rows by X n starting the numbering of the layers in the vicinity of the dislocation centre: X n = nb, where n = 0, ± 1, ± 2,. .. The total energy of misfit of the crystal with one single dislocation is
E=
(95) n = -
00
n = -
00
It follows from eq. (92) that
~ ) ~ (x
_
~2
+ A2·
(96)
Substituting eq. (96) in eq. (95) and carrying out the summation over n by an aid of Poisson's summation formula, we obtain for the energy per unit length of the dislocation in the first approximation (A » b):
= Db + 2Db e- 2 n A/ b cos (2nX/b).
(97)
It is seen that the dislocation energy is indeed a periodic function of the dislocation position. By using eq. (97) we may 0 btain the force on the dislocation produced by the crystal structure
Thus, the Peierls equation takes the form
foo
rU~)
p<;j(x) du = -bfll
= fll b2/TC sin" (nu(x)/b).
E
III = Il b/ na(3 - 2v).
p
Jo
sin' (nu(x)/b) = cos" (tan -1 x
Thus, at large distances from the dislocation core, the relative displacement is u(x)/b = D/2fll nax.
rU~)
Vex) = -b
(89)
x
77
It is easy to estimate the critical shear stress which is required in the Peierls model to start the movement ofthe dislocation in the glide plane. This was estimated by Nabarro [30] by using the following consideration. The stress p<;}(x) produces the potential energy of any atom row
(88)
The solution of eq. (88) should satisfy the boundary condition of eq. (86). Now we find the relation of the coefficient fll to the elastic modulus of the crystal. It is clear that far from the dislocation core (x » b), where the displacements u(x) are small, the solution of eq. (88) should coincide with the solution given by elasticity theory Ceq. (22)J if one takes y = 0 and oUx/oy = u(x)/a. It follows from eq. (88) that for x ---+XJ, when u « b, we have 2nfll u(x) = _ 1
Crystal dislocations and the theory of elasticity
§5.5
oE = -Zub '. - e- 2 n A/ sm (2nX/b). oX 1 - v
F = - -
b
(98)
The formula of eq. (98) determines the so-called Peierls-Nabarro force. The maximum value of this force Fm determines that shear stress as = Fm/b which should be applied to the crystal in order to initiate the movement of the dislocation in its glide plane. To estimate o"s, we take A = a and a = ~b. It appears then that as '" 10- 4 fl. If a = b, then as'" 10 - 2 fl. These are the values that were given above in the discussion of dislocation glide.
(93)
The value A may be conditionally called the half-width of the dislocation. The conditionality of this concept is connected with a slow spatial change of the displacement u(x), and that makes the estimation of the dislocation width rather strongly dependent on the particular form of the periodic function p<;}(x). The dislocation half-width as a function of the magnitude of Poisson's ratio may take values in the range from A = ~a (for v = 0) to A = a (for v = ~).
5.6. Dislocation sources To ensure a considerable plastic deformation, there should exist some mechanisms of dislocation multiplication in the crystal, i.e. some dislocation sources should function. One of the simplest schemes of a continuously operating source of dislocations was suggested by Frank and Read [31]. We consider a straight dislocation segment fixed at two points A and B. This may
cu.i
A. M. Kosevich
78
(b)
Fig. 31 A dislocation segment fixed at two points A and B. (a) A and B are nodes of the dislocation network. (b) A and B are bends on the same dislocation.
be a segment between two nodes of the dislocation network (fig. 3la) or between two bends in the same dislocation (fig. 31b). We assume that a uniform applied-stress has only that component which creates a force in the glide plane of the element AB and does not create forces in the glide planes of other adjacent dislocation segments. Then, in the glide plane of the element AB there is a force/which is given by eq. (62) and is directed normal to any element of the dislocation AB. If this force exceeds the retarding force in the crystal, the dislocation line AB begins to move, remaining fixed at the points A and B. As a result it will bend in the form of a certain loop which connects the points A and B. The shape of this loop can be found by using a simplified mathematical description, the so-called line tension approximation (see sect. 3.4.). If the line tension To is independent of the shape of the loop and of the orientation of the element of the dislocation line, the equilibrium form of the loop is a circular arc of a radius R (fig. 32) given by the condition
I
To
>
If + F res '
(99)
where F res is the static force of the crystal resistance acting on the dislocation. The line tension is approximately To ~ ub", It is clear that there is a certain critical equilibrium radius of the arc R o , where 2R o is the distance between the points A and B. For small values of/when
/ <
To
/0 = IF + r.; o
Fig. 32 A stable circular arc of dislocation AB has a radius R > R o' where R o is the critical dislocation radius.
Crystal dislocations and the theory ofelasticity
§5.6
79
exists always an equilibrium dislocation loop of radius R > R o . If/ > /0' the loop has no equilibrium form. In this case, the dislocation will move, expanding freely in the glide plane. The successive positions of such a loop in the glide plane are shown in fig. 33. The original line 1 bulges to a loop 2, then to 3. At 4 two elements of the loop are about to meet. They annihilate, and the next configuration is a closed loop 5 together with a short segment of the dislocation joining A and B. The loop 5 now expands freely as a single dislocation line. The short segment again begins to pass through the successive configurations 2, 3, 4, ... The dislocation segment AB, which can give rise to a succession of closed loops of dislocations, is called a Frank-Read source.
Fig. 33 The Frank-Read source. A segment AB of the dislocation line is held fixed atA and B. Under an applied stress the segment AB moves from 1 to 2, then to 3, 4 and finally to 5.
Let us write / = ba, wherec is an external shear stress. Then the critical shear strength (J 0 which is required to operate a Frank-Read source of length 2R o is given by (J
o
F res = /0 - = -To - +-.
b
bRo
b
(100)
The relationship of eq. (100) is obtained by using the line tension approximation. However, Indenbom and Dubnova [2lJ have shown that this approximation does not provide a correct description of the dislocation configuration near the nodes in the network or bends at the points A and B in fig. 31. The exact calculation [32J yields a value of the critical shear stress which exceeds that obtained in the line tension approximation. If (J > (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 <5x = br0 in an external elastic field. Then the strains ekk in eq. (225) would change by bekk = (oekkloxJ<5x i , and the energy change may be represented as <5Eint = - F bX, where the force F is equal to (226) where Po is the hydrostatic pressure in the crystal. This is the force with which the elastically deformed crystal acts on the defect. Thus, the centre of dilatation in an isotropic medium (in a cubic crystal, also) experiences a force proportional to the gradient of the hydrostatic pressure. Equations (225) and (226) allow us to describe the elastic interaction of the point defect with a dislocation. Imagine that in the crystal there is a dislocation which has a length large compared with the mean distance between point defects. Then we may ask the equilibrium distribution of centres of dilatation in the dislocation stress field. We assume that point defects form a dilute solution. Then the equilibrium concentration C of point defect is given in terms of the concentration Co far from the dislocation by c = Co exp ( - EintlkT),
(227)
where T is the crystal temperature. According to eqs. (225) and (227) the distribution of point defects should be sharply nonuniform and essentially dependent on the orientation and on the type of dislocation. To illustrate this dependence we consider an equilibrium distribution of defects near an edge dislocation. It follows from eqs. (26), (225) and Hooke's law that c=
Co
exp [ - A(bflQolkT) sin Olr] ,
A > 0,
where r is the distance from the dislocation axis (r » b) and the polar angle 0 is taken from the direction of the Burgers vector of the dislocation. The positive constant A has the order of magnitude of unity. Bearing in mind that Q o > 0 for interstitial atoms and Q o < 0 for vacancies we can conclude that near the dislocation there are always regions of heightened concentration of defects of one type or another. Thus, the elastic interaction of point defects with the dislocation leads to the formation of "clouds" of point defects (Cottrell atmosphere) near the latter. It is easy to verify that the excess concentration of vacancies appears above the slip plane of the edge dislocation (on the side of an "extra half-plane") and the excess concentration of interstitial atoms occurs under the slip plane.
Crystal dislocations and the theory of elasticity
§9.2
119
9.2. Dynamical retardation of a dislocation by heavy impurities At the beginning of the preceding section it was noted that point defects play the role of sources of internal stress. However, the point defects can be considered at the same time as local inhomogeneities in the crystal [40]. If a point defect is formed by an impurity atom, it leads to a local change of the crystal mass density as well as a local disturbance of the regular atomic arrangement in its vicinity. And so the presence of the impurity significantly changes the dynamics of the crystal lattice : in particular, it profoundly alters the frequency spectrum of the normal modes of vibration of the crystal. Under certain conditions the transformation of the spectrum can involve the low frequency region corresponding to long-wave vibrations of the crystal. The transformation of the spectrum may be described macroscopically in terms of dispersion of the elastic moduli caused by impurities. This dispersion leads to the damping of elastic waves in the crystal and the slowing down of a moving dislocation. We consider a disordered solid solution of low concentration c(c « I), produced by interstitial impurities in the regular crystal lattice. Let rno be the mass of the crystal atom and M be the mass of the impurity atom. We are interested in the singularities introduced by the impurities in the long-wave deformations of the crystal, relative to which the crystal can be regarded as a continuous medium, and the impurities and the perturbations caused by their introduction can be regarded as pointlike. In addition, wishing to obtain the results in a closed form, we must recognize that the scattering of elastic waves and the deceleration of dislocations are described by relatively simple formulae only in the isotropic approximation. Therefore, it is reasonable to confine ourselves to the simplest crystal model, namely a continuous isotropic medium. In this approximation, the perturbation of the atomic force constants can be characterized by a single parameter U0 describing the "strength" of the perturbation [77]. For elastic oscillations whose wavelengths greatly exceed the average distance between impurities, we can obtain the following two dispersion equations [77J: s2k
2
= w2
-
wi{cA(w 2)/[wi
+
A(w 2 ) (l
+
iw/w 2 )J} ,
(228)
one of which determines the dispersion law for longitudinal waves (s = ce) and the other for transverse waves (s = ct ) in a crystal with interstitial impurities. Here, k is the wave vector, and (229) The constants WI and W 2 are determined by the velocities of the sound waves and have the order of magnitude of the Debye frequency. It follows from eq. (229) that the "bare" self-frequency of the impurity w s ' where
w; =
Uo(rnoIM),
(230)
may be taken as the only parameter of the isotropic model. The fact that the presence of impurities does not lead to "mixing" of the longitudinal and transverse waves in the crystal is a consequence of the isotropy of the perturbation potential model and the linear approximation in the concentration c.
A. M. Kosevich
120
Ch.l
We note that for substitutional impurities, a dispersion equation similar to eq. (228) was obtained by a somewhat different method by Slutskin and Sergeeva [78] with a more general form of the perturbation potential. In the final analysis we are interested in the damping of a straight dislocation moving fast in a crystal with a constant velocity V(c(a s / Jl)1/2 « V« s). In the isotropic approximation, this force per unit dislocation length is determined by eq. (206). The value of the damping force is determined completely by the dispersion of the elastic moduli (see sect. 8.4.). The explicit form for the dependence of the wave vector on the frequency eq. (228) enables us to investigate in detail the modulus dispersion due to the impurities. The singularities in the propagation of elastic waves in a crystal with impurities appear in the vicinity of the renormalized natural frequency co,, if this frequency lies in the low-frequency end of the spectrum. The frequency W r is equal to W r
=
W
1+ 8 (
M ( 2 ) - 1/2
3=~
mo wi
w;
~ (mo/ 3M )1/2w 1 "" (mo/M)1/2 wD,
(232)
where W D is the Debye frequency. Let the following conditions C
2/ 3
be satisfied simultaneously. Under these conditions, as shown by Kosevich and Natsik [77, 79], the retardation force of a dislocation by impurities has the following dependence on the velocity V. If the velocity of the dislocations is SUfficiently small, V « aoi., then the retardation force increases according to the law (233) where the order of magnitude of the coefficient a? is
(M)2 (ak o)4 """ .. C.. (M.)2 bu, -_. b Jl--m m
IX 2 "" c. -.-
o
and in the case of the screw dislocation it is
M)
V os = - 1 ( c - b 3 w s ' 4n mo
(237)
Thus, the deceleration force of the dislocation, as a function of the dislocation velocity, has a sharply nonmonotonic character if the crystal contains impurities with quasi-local frequencies. If the impurity does not produce any quasi-local vibration, then the dependence of the deceleration force on the dislocation velocity is described by eq. (233) for all velocities that admit of a macroscopic analysis. It must be borne in mind, however, that the coefficient entering into this formula is defined very roughly, since it contains a factor k6, that is, a parameter whose order ofmagnitude can only be estimated. As regards eqs. (235) and (236), they are accurate in this sense, that they contain only the macroscopic characteristics of the medium in the region of their applicability. 9.3. Speed of climb of an edge dislocation
wr.
c « molM «
121
(231)
,
where is determined by eq. (230). Then it follows from eq, (231) that We assume that V o » «i,
Crystal dislocations and the theory of elasticity
§9.2
S3
o
(234)
S3
and ko(ak o "" 1) is a maximum wave number introduced to cut off the integration in eq. (206) (see sect. 8.4.). For rather large velocities of the dislocation, V » tua., the retardation is inversely proportional to the velocity f(V) = -bJl(Vo/V),
(235)
where in the case of the edge dislocation the parameter V o is [79] cf ) ,
cj
(236)
The presence of the field of strains and stresses round the dislocation leads to its elastic interaction with point defects. However, there is an interaction between the dislocation and point defects of another nature, which we call a diffusional interaction. In the analysis of dislocation motion (sect. 4.2.) it was shown that dislocation climb involves an inelastic change of the volume of the medium. It follows from eq. (42) that this inelastic increase of the relative volume per unit time q(r) = de~k/dt is equal to q(r) = V(b x t)
<5(~),
(238)
where V is the dislocation velocity. This same formula follows from the physical meaning of the quantity io = ikk and the relation of eq. (162). Up to now we have considered the situations when nonconservative motion of the dislocation initiates inelastic change of volume q. However, the relation of eq. (238) may be treated in another way, namely, if the change of volume occurs on the dislocation line, then the dislocation has to climb with a velocity determined by eq. (238). The local inelastic change of volume takes place, for example, when a macroscopic quantity of interstitial atoms or vacancies is absorbed by the dislocation ("condensation" of point defects occurs on the dislocation). If the "condensation" of point defects is realized by their diffusional supply to the dislocation core, then the inelastic increase of the crystal volume along the dislocation line can be determined by (239) where I is the difference between diffusional flows of vacancies I; and interstitial atoms I, per unit dislocation length. Interstitial atoms absorbed on the dislocation axis become "regular" atoms of the lattice, therefore each interstitial atom brings the atomic volume a 3 in the process of "condensation" (the vacancy takes away the same volume). This peculiarity is taken into account in eq. (239). In this approach the problem oLille nonconservative motion of the dislocation
A. M. Kosevich
122
Ch.l
reduces to the calculation of bulk diffusional flows of point defects. The calculation of the flow I onto the dislocation implies usually the following mechanism of "condensation" or "evaporation" of point defects. The dislocation core is surrounded by a tube of radius roCr 0 '" b '" a) and it is assumed that the point defect is absorbed (or emitted) when it reaches and intersects the surface of this tube. Further, for the sake of simplicity we shall consider only the vacancy flow, taking I = Iv' Comparing eqs. (238) and (239) we can conclude that (240) An evident transformation of the left-hand part of eq. (240) yields V(b x t) = (b x t)Vn = b1.Vn'
Crystal dislocations and the theory of elasticity
8J1(c) {yc {yJ1vlr= 00 = -8- {yc = kT-, c
Thus, the nonzero flow of point defects towards the axis of the dislocation causes climb of the latter with the velocity of eq. (242). The diffusional flow to the dislocation may be naturally due to the supersaturation of point defects in the bulk of the crystal. However, this flow may be also induced by a force acting on the dislocation in the direction normal to the glide plane. Let the dislocation be in a field of stresses creating a force F on unit length of the dislocation. Then in the climb of the dislocation work {YR is done which is determined by the quantity FVn per unit time. First we write this work related to a single vacancy absorbed on the dislocation in the case of a pure edge straight dislocation:
= -
coDy a 3 k T grad J1v'
Here the x-axis is directed along the Burgers vector of the dislocation and P~x is the corresponding component of the stress tensor-deviator. Let us assume that the applied force is weak, and the supersaturation of vacancies is low. Then the dislocation climbs slowly. Suppose that in the process of motion local equilibrium of point defects is established near the dislocation axis with respect to the transition of the defects through the surface of the dislocation tube. This supposition allows us to avoid a discussion of the kinetics of climb, i.e. the kinetics of the accumulation ofthe point defects on a dislocation [80J. The equilibrium boundary condition coming from eq. (243) for the change of the chemical potential of the vacancies J1v at the dislocation tube surface has the form (244)
(245)
(246)
where D; is the coefficient of vacancy diffusion. Now consider a single straight dislocation parallel to the z-axis. This dislocation climbs along the y-axis with the speed Vy = (a 3 Ib)I. We find the chemical potential J1v satisfying the conditions of eqs. (244) and (245), and calculate the total vacancy flow I per unit dislocation length by using eq. (246). Then the speed with which the dislocation climbs is
v
=
y
2nc oD v (a3p~x {yc) b log (Llr o) kT - Co '
(247)
where L is the dislocation length or the crystal size. Let us assume that the concentration of vacancies in the crystal is equilibrium ({yc = 0) when the stationary diffusion flows are determined. Substituting {yc = 0 in eq. (247) we find V = y
(243)
Co
where {yc is the supersaturation of vacancies, and Co is the equilibrium concentration of vacancies in the absence of externalloads*. Since the chemical potential in the stationary case is a harmonic function** satisfying the conditions of eqs. (244) and (245), it can be readily obtained in a standard manner. The density of the diffusion vacancy flow is determined by the formula j
(242)
123
The presence of a low supersaturation of vacancies far from the dislocation is taken into account by the following condition at infinity
(241)
where Vn is the component of the dislocation velocity normal to its glide plane, and b1. is the magnitude of the component of the Burgers vector perpendicular to the dislocation line at a given point (bi = (b x t)2), i.e. the magnitude of its "edge" component. It follows directly from eqs. (240) and (241) that Vn = (a 3 I l bi Hb x t).
§9.3
2nc oD v (a3p~x) b log (Llr 0) kT
If there are a great number of similar parallel edge dislocations in the sample, their climb causes the following rate of the average plastic strain:
eP = -
xx
3P (a . ~x), o) kT
b V = 2nPdcODy Pd log (Llr
Y
(248)
where Pd is the dislocation density (the number of dislocations per unit area on the x-y-plane). * It should be noted that bc is the effective average supersaturation which is established within the sample (far from its surface) in a volume including a large number of dislocations. ** Vershok and Roitburd [8lJ have determined the limits of applicability of the assumption that the diffusion problem under consideration can be reduced to the solving of Laplace's equation provided the concentrations of vacancies are local equilibrium ones near the dislocation and far from it, that is the limits of validity of the assumption that the chemical potential u; can be considered as a harmonic function, which satisfies the conditions of eqs. (244) and (245).
124
Ch.1
A. M. Koseoich
Thus, even rather weak stresses can initiate dislocation-diffusion creep of the single crystal. The physical scheme of the plastic deformation of the sample with such a type of creep is clear from fig. 48, where arrows at dislocations show the direction of the dislocation climb. It is also clear that if only edge dislocations with parallel glide planes (as sketched in the figure) were present in the crystal, the mechanism of creep described by eq. (248) would soon cease functioning. In fact, the process of the dislocation climb considered is connected with a change of the point defect concentration in the bulk of the crystal. It is easily seen that the rate of change of the vacancy supersaturation bC has a sign coinciding with that of P~x' Therefore, the moment of time necessarily comes at which
Crystal dislocations and the theory ofelasticity
§9.3
Therefore, the vacancy diffusion induces a dislocation of the second set to climb with the following speed: V _ x -
bC)
2nc OD y (a3p~x b log (Llr 0) kT - Co .
(250)
The evident condition of overall balance of the transported matter, under which = 0 and steady-state creep takes place, reduces to the condition
dc/dt
p~1)i,
+
p~2) 12
= 0,
or
bC) + Pd (a3p~y _ bC) _ kT Co - 0,
(l)(a3p~X
(249)
and the plastic flow of the crystal stops (Vy = 0) as the result of the creation of the vacancy supersaturation bc defined by eq. (249).
125
(2)
kT
Pd
(251)
Co
where p~1), p~2) and Ii' 12 are dislocation densities and vacancy fluxes per unit length of the dislocations for the first and the second sets, respectively. The stationary supersaturation ofvacancies in the crystal is obtained from eq. (251): 3
b - co(a /k T ) (1)' C - p~1) + p~2) (Pd Pxx
~xx
(2)
,
+ Pd p yy ) .
(252)
We take eq. (252) into account in eqs. (247) and (250) for the dislocation velocities and calculate the average rate of the plastic strain in the crystal, i.e. the creep rate eP
Fig. 48 The climb of a set of parallel edge dislocations with parallel Burgers vectors.
xx .p
The situation changes if in the crystal there are two sets of parallel dislocations which have perpendicular glide planes (fig. 49). In this case the calculated growth of extra half-planes would be due to the "dissolution" of extra half-planes perpendicular to them, associated with the other set of dislocations. It is easy to obtain the supersaturation bc at which the stationary process of "pumping over" the matter from one set of dislocations to the other will take place. Let the first set be that set of dislocations which has its Burgers vectors directed along the x-axis and the second set that which has its Burgers vectors along the j-axis. Each dislocation of the second set climbs along the x-axis with a speed V x = -(a 3 / b)I . 1-
t Fig. 49
--I _1
t -....
-p(1)bV d Y
=
2nc D 0
y
p(1)p(2) d d
log (Llr 0) p~1)
.p
eyy = -ex x ,
+
p~2)
a 3 (p
xx-P yy kT
)
,
(253)
The steady-state creep of eq. (253) naturally causes a pure shear deformation of a sample. The steady-state diffusional creep described by eq. (253) occurs when two conditions are satisfied. Firstly, in the process of creep the dislocations have their straight line forms unchanged, i.e. they are not fixed at the nodes of the dislocation network and are not fixed at certain points by impurities. Secondly, no new dislocations are generated. Nabarro [82J has considered the steady-state diffusional creep of the crystal for the case when dislocations form a certain equilibrium dislocation network and their density increases due to Bardeen-Herring sources. 9.4. The climb of a helical dislocation
1-_I....
=
:)pxx
Dislocations of two types with perpendicular glide planes are acted on by the stress directions of climb are indicated by arrows.
Pxx'
The
Note the peculiar feature of eq. (242) for the case of a pure screw dislocation. For the screw dislocation b1. = 0 and for any nonzero flow I the climb speed of the screw dislocation given byeq. (242) turns into infinity and its direction becomes indefinite. This peculiarity has the following physical meaning: the screw dislocation is absolutely unstable with respect to the climb due to the condensation of point defects. Even a small flow of point defects onto the dislocation leads to a displacement of its screw segment which results in a bending of the dislocation line and the formation of an
A. M. Kosevich
126
Ch.1
edge component of the Burgers vector. This bending of the dislocation causes the appearance of line tension and prevents further motion of the segment. It is easily understood that in an isotropic medium the initially isolated straight line screw dislocation acquires the shape of a helix after the loss of stability under the effect of a diffusion flow which is uniform along its length. The axis of the helix coincides with the primary direction of the dislocation (see sect. 4.2.). The glide surface of the helical dislocation is a circular cylinder on which the dislocation is wound (fig. 50). Therefore, any dislocation element climbs parallel to the normal n (fig. 50), i.e. its speed is directed perpendicular to the axis of the helix. And so, under uniform conditions the climb of such a dislocation cannot change a constant pitch of the helix 2nh (fig. 50) and leads only to an increase of the helix radius R. The pitch or the number of turns of a helix with a given length L is determined by random initial deviations of the dislocation from a straight line shape at the moment when it ceases to be stable. For a very long helix (L » R) fixed at its ends, the dislocation glide along the axis may be neglected and the quantity h may be taken as a constant. Then the change of the shape of the helix by climb would reduce to the change of its radius R and the speed of eq. (242) would be Vn = R.
§94
Crystal dislocations and the theory of elasticity
tx
= -
tz
=
R(R 2
h(R 2
127
+ h2 ) -1/2 sin cp,
+ h2) - 1/2
(256)
where tp is the angle taken in the x-j-plane from the x-axis to the direction of the vector t. According to eq. (254) the change in the chemical potential of vacancies u; on the dislocation tube surface is (257) Note that only the total vacancy flow per unit length of the dislocation tube I is used in the calculation of the dislocation climb speed of eq. (242). Then the problem of interest to us, to find the flow onto dislocation loops of symmetrical shapes in the uniform field of stresses (a* = constant, I = constant), is equivalent to the problem of finding the line density of electrical charge along the corresponding conducting line. Using this electro-diffusional analogy, it is easy to obtain the following expression for the total vacancy flow v C 1= 4nc oD a3
{bC _ a
3a*}
,
(258)
k.T
Co
where C is the electrical capacity per unit length of the conducting line. As i; and t y are functions of position on the helix, the assumed constancy a * along the dislocation line is possible only in an external field with Pxz = Pyz = O. As follows directly from eq. (255), in this field ao(R)
Fig. 50 The helical dislocation. The tangent vector t, Burgers vector b and unit vector n normal to the slip plane form a right-hand triplet of vectors.
However, in deriving I we should now take into account the fact that the helical dislocation is neither purely of edge type nor straight. Therefore, instead of eq. (243) one should write (254) where a; == hE = b - (th)tis the edge component of the Burgers vector. We include the static self-action force of the helical dislocation F 0 as a separate component of the effective stress a *. The value of F o can easily be calculated if the self-energy of the helical dislocation E H as a function of the radius R (for a fixed h) is known. The necessary expressions for the function E H = EH(R) are given by de Wit [45] and we shall not write them here. We only note that for the helical dislocation with its axis coincident with the z-axis (fig. 50) we have E bk = b 2(pzz - tztmPmz , b2.l = R 2b 2j(R 2 + h 2), bmPmk (255) I
I
')
=
(259)
where Fn(R) is the projection of the static self-action force of the helical dislocation on the normal to the glide surface. This quantity depends naturally in a complicated manner on the helix radius: F; = FiR). If the vector n is directed from the helix axis (as in fig. 50), then F; < O. Substituting eqs. (259) and (258) into eq. (242) we obtain the rate of change of the helix radius: dR
dt
C(R)
{a
3 I
= 4nc oDv b.l(R) kT [pzz - aoCR)] -
bC}
Co
'
(260)
where C(R) is the capacity of unit length of the helix and the dependence b.l = b.l(R) is given by a formula from eq. (255). The sign of the expression in the curly brackets in the right-hand side of eq. (260) shows if the helix radius increases (R > 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--<
X X X X (b)
(a)
Fig. 51 The formation of a vacancy prismatic dislocation. (a) A sheet of vacant sites in a perfect crystal. (b) The plane accumulation of vacancies may collapse to form a prismatic dislocation loop.
(263)
.L
As the vector Vn lies always in the plane parallel to the x-y-plane eq. (263) corresponds to the displacement of the helix as a whole along the x-axis with a velocity Vx
= const a 3p y z/ k T.
(264)
To obtain the dependence of the constant factor in eq. (264) on the helix radius R and the parameters co, D v ' one needs a more detailed consideration of the problem which was made by Roitburd [83J. Roitburd [83J has also considered a stationary nonconservative motion of a dislocation with an arbitrary shape initiated by the bulk diffusion of vacancies. 9.5. The formation and the growth of prismatic dislocation loops The process of the formation of prismatic dislocation loops by specific accumulation of point defects is well known at the present time. Thus, plane accumulations of vacancies may collapse after quenching, and this leads to the formation of dislocation loops limiting the "removed" parts of atomic planes. The process of the formation of such a prismatic dislocation is illustrated by fig. 51. Dislocations of this type will be called vacancy dislocation loops. Under the influence of radiation an excess of interstitial atoms occurs and these atoms may pile up into "pancakes" with the
* If on the dislocation J.1v ex cos tp, then a stationary linear flow of vacancies arises along the dislocation line, and it should necessarily be taken into account. But it is easy to see that the contribution of this flow to I is also accompanied by a factor cos tp.
Fig. 52 Section of a loop of a prismatic dislocation referred to as an interstitial dislocation.
To facilitate quantitative analysis of the processes of growth or dissolution of a loop we assume that the dislocation is complete and that the Burgers vector is perpendicular to the plane of the dislocation loop. We shall also assume that the medium is isotropic and that the equilibrium shape of the loop is a circle. The glide surface of such a dislocation is a circular cylinder, and we suppose that the normal to this cylinder n is directed away from its axis. The size of a circular dislocation is characterized only by its radius R, whose value varies with the number of vacancies absorbed or emitted by the dislocation. The rate of change of the radius of the loop dR/dt = V n is determined by the relationship of eq. (242), where b..L = band b x t = bn for the vacancy loop but b x t = - bn for the interstitial loop. The flow of vacancies is given by eq. (258), where we have to substitute the magnitude C = [2 log (8R/r o)J 1 "-' [2 log (R/ro)J - 1 which determines the electrical capacity
Ch.1
A. M. Koseoich
130
per unit length of a circular wire. Finally, we obtain the following rates of growth of the dislocation loops
(a
27rc oD y
dRy dt
bC)
3a;.
b log (R/ro)kT -
dR j
dt =
2nc OD
y
b log (R/r o)
(a
Co
'
bC)
3a;
kT -
Co
(6 ) 2 5
.
The effective stresses a; and at include the line tension of the dislocation which is calculated in the usual way with an aid of eq. (153). Let us consider a crystal under a unifonn uniaxial load (we denote the corresponding component of the stress tensor by a). Let () be the angle between the Burgers vector (or a vector normal to the plane of the loop) and the axis of the external load. Then* 1 = a(cos 2 () - -) +
a*
3
y
= a(cos
a:J'
2
e - -) 1
3
1 R -log-, 4n(1 - v) R r0
v
J.1b 1 R log-· 4n( I - v) R r0
(266)
We can usually obtain on the basis of eqs. (265) and (266) the rates of change of the radii of the dislocation loops [84, 85J: 2nc oD y [ b log (R/r ) o
dRy dt -dR
dt
j
_
-
2
K(COS
() -
1
bc
3) -
Co
difference in the corresponding equilibrium concentrations at the dislocation and far from it. The former (at the dislocation) is governed essentially by the orientation of the loop of the given type in the external stress field. The latter (far from the dislocation) is connected with the overall balance of point defects, both situated within the crystal (in the solution) and those entering into the prismatic dislocation loops. It is obvious that at a fixed external load the different loops are under different energy conditions, and the diffusion flows will contribute to the growth of the "favourably located" dislocation loops at the expense of the others. For those orientations for which cos? () < ! + bC/KC o, all the interstitial loops must decrease (dR)dt < 0), and ultimately they are completely "dissolved" in the crystal. As to the vacancy loops, some of them are also dissolved, and those having radii larger than a certain critical value R~r increase in size. The critical radius is determined by the obvious relation A(Rcr)
J.1b
+
l J'
b A(R) R
(267)
2nc O D y [" K ( cos 2 e - 31) - -bc - A (R) bJ, b log (R/r o ) Co R
(268)
where a 3 J.1 R A(R) = 4n(1 _ v) kT log "o
I
* If the only non-zero component of the tensor Pik is pzz = (J then pin = niP;knk er(cos2 e - j), where e is the angle between the unit vector n and the z-axis.
jpu =
R" y
bc _ C
A(R~r) ~ = 1
K(COS 2 () _
0
1.) for
bc >
3
C
K(COS 2
e _ 1). 3
0
(269)
Rfr
bc Co
+
K(COS 2
e _ 1.) 3
for
bc < Co
K(COS 2 () _
1.). 3
(270)
Thus, a "pumping over" of matter will take place from certain atomic planes to others, and will give rise to plastic deformation of the sample. This diffusiondislocation mechanism of steady-state crystal flow was analyzed by Kosevich, Saralidze and Slezov [86]. If there is no external load (K = 0), then eqs. (267) and (268) become simpler. Thus, in particular, the rate of growth of the vacancy loop is dt
= niPiknk -
~=
Analogously, in the angle interval cos' () > ! + 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.
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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)
where
Zn = Xl
+ Pnx2
~nd P« are the roots of the sextic equation obtained by substituting eqs. (59) and (60) mto eqs. (57) and (58). To complete the analysis it is only necessary to evaluate either B; or C; from the conditions in eq. (11), since we have Y = Cn = S24 - Pn(S25 + S46) + P;;(S14 + S56) - P~S15 n B; S44 - 2PnS45 + P;;S55
4. Two-dimensional problems 4.1. The stress function method In looking for simple analytic solutions it turns out by eq. (6) to be more convenient to solve the compatibility equations than the equilibrium equation (9). For a straight dislocation along X 3 the compatibility relations reduce to the two equations 2e 2e 2e a 11 + a 22 = a 12 , (55) ax~ aXI aX 1 aX2
ae 13 aX2
_
(J22 = fJlBn/z n,
(61)
(J23 = -fJlCn/zn·
(62)
Finally the energy factor (eq. (51)) has the form 3
ae 23 = O. aX1
(56)
E(8) = fJl
I [( -b l P2 + b2)B n -
n= 1
We now replace ei j by (Jij according to eq. (8) and derive aij from two stress functions Fand
(J23 = The resulting forms of eqs. (55) and (56) may be written
The. advantages of this solution over the alternative derived from the equilibrium equation [17, 18J are two-fold. In the first place there are only two equations instead of three, and secondly it is only necessary to evaluate one set of quantities (B or C ) in addition to the sextic equation. Then, for example, n n
b 3Cnl
(63)
4.2. General simplifications
I~ evaluating t~~ stress field of an infinite straight dislocation we have in general eighteen quantities to consider, six for each separate Burgers vector component. Three of these, say the three contributions to a 33' are directly obtainable from the other fifteen by the condition e 33 = O. We now demonstrate that it is only necessary to find three of the remaining fifteen quantities by first forming a dislocation with Burgers vector [b 1 , b2 , b 3 J with a radial cut procedure. Let us describe the process in both Cartesian and cylindrical polar coordinates.
J. W. Steeds and J. R. Willis
156
Cb.2
We write the energy factor of eq. (63) as a sum over components referred to Cartesian coordinates E((J)
= Kijbib j.
Then it follows that [19]
(Jr8
=
Kub i cos (J + K 2ibi sin (J 2nr·· ,
(J
_ K 2ibi cos (J - K 2i bi sin (J 2nr
88 -
U sing the relationship between Cartesian and cylindrical polar stress field components we obtain [20]
(J 11 (J 22
e, = (J 12 tan e + K 2ibd 2nr cos e,
= (J 12 cot (J
-
K libd2nr sin
(J32 = cr 13 tan e + K3ibd2nr cos e. (64) N ext we consider three interactions between parallel dislocations which would vanish according to isotropic elasticity. For equal action and reaction between a screw and the edge dislocation with Burgers vector [b 1 , 0, 0] we obtain
(66) Further the interaction between the two edges themselves gives
(J12(b 1)b2 = cr 11(b 2)b 1,
(J22(b 1)b2 = cr 12(b 2)b1.
157
performed as a final step in the analysis. For similar reasons the stress function approach, which greatly simplifies the algebra, has proved particularly powerful for finding analytical solutions. Analytic results have now been obtained for many practical cases. Complete solutions exist for three distinct cases and partial solutions for a number of others. The first completely solved case has the minimum symmetry requirement that the dislocation lies along a diad axis with another diad perpendicular. A complete list of examples of this type is given in table 1. Diad here is used in the sense of a two-fold axis for elastic properties. Since elasticity is a property which adds a centre of symmetry, mirror planes in the crystal symmetry elements are also included in the description. The second completely solved case is that of a dislocation along a triad axis with a diad perpendicular. This is a simple example of the rather general third case, that of a dislocation with a diad axis perpendicular to it. When this diad axis is the slip plane normal all possible glide dislocations are covered by the analysis. A summary of examples where analytic solutions exist is presented in table 2. To give some idea of the simplicity of the results which emerge we give a few examples. Let us first consider the case of a screw dislocation along a diad axis with another diad perpendicular. Then the displacement field is
(65) The interaction between the same screw and the edge [0, b2 , 0] yields
Dislocations in anisotropic media
§4.3
u,
= :"
tan-I [t;~I~J
(68)
where A is an anisotropy factor whose exact definition depends on the crystal system and the choice of coordinates. In the isotropic limit A -* 1. When, however, the screw dislocation lies along a triad axis with a diad perpendicular the component of the displacement field parallel to the dislocation line is given by
(67)
Equations (64)-(67) are relationships which allow all possible stress field components to be generated by starting from anyone component of the stress field of a pure edge dislocation, e.g, (J 12(b 2 ) , one component of the stress field of a pure screw dislocation, e.g. (J 13(b 3) and one mixed stress field, e.g. a 12(b3)·
_ b -1 ( tan 3e ) 6n tan (1 _ (j)1/2 '
where (j is another anisotropic parameter which, in terms of reduced compliances, is defined by (j
4.3. Effects of crystal symmetry It is fortunate that very often during plastic deformation crystals slip on planes and in directions of symmetry. The explanation of this behaviour lies largely in the dislocation core beyond the reach of elastic theory, but the elastic strain fields are greatly simplified as a result. In some cases the slip behaviour is itself dominated by elastic energy considerations (see sect. 5.5.). In order to make use of the slip symmetry it is important to refer the problem to carefully chosen coordinates. Although no general rules have been established it seems to be preferable to make the dislocation direction one axis (normally x ,) of the coordinate system. The other two orthogonal axes should be chosen to lie along two, four or six-fold axes when possible. In exploiting the symmetry elements orthogonal to the dislocation it may be that the slip plane is inclined to the chosen axes. However, the first priority is to simplify the algebraic steps which lead to an analytic result: then coordinate transformation may easily be
(69)
U3 -
(j
-*
= Sis/S 11 S44'
°
in the isotropic limit. Table 1 The six cases when screw and edge components of a dislocation may be treated separately Dislocation axis
Perpendicular axis
diad tetrad diad tetrad hexad
diad diad tetrad tetrad hexad
J. W. Steeds and J. R. Willis
158
Ch.2
Table 2 Examples of analytic solutions [6J Crystal system
Slip system
Analytic cases
Cubic
{11O} (110)
Any glide dislocation
Ionic crystals of high polarisability (NaCl, AgCl, MgO), non stoichiometric spinels Fluorite structures (VO z, CaF z) and Any glide dislocation ionic crystals which are almost covalent (PbS, PbTe) Screw, 30°, 60° and edge f.c.c. metals and alloys, diamond structures, sphalerite, ionic carbides, cubic Laves phases CsCl structures (AuCd, NiAl) Any glide dislocation b.c.c. metals, f3 phases of alloys with Any glide dislocation CsCl structure (AgMg, CuZn) Screw, 39°, 58tO and edge Some b.c.c. metals (Nb, W, Fe, Mo) Screw, 43°,56°, 75°, 97°, Some b.c.c. metals 118° and 143° dislocations Any hexagonal material Any dislocation White tin, ric, Any glide dislocation TiO white tin Screw, edge and one z, intermediate Sb, Bi, exAl z0 3 , BizTe 3 Screw, 30° and edge Te, exAl z0 3 Screw, edge and one intermediate Ga Any glide dislocation exV Any glide dislocation
{001} (110)
{Ill} (110)
{110} (001) {110} (Ill) {211} (Ill) {312} (Ill) Hexagonal Tetragonal
All systems
Trigonal
{0001} (1120) {1I00} (1120)
Orthorhombic
(001) [OIOJ (010) [100J
Examples
PIO} [OOlJ {101} (lOI)
Dislocations in anisotropic media
§4.3
159
tion. However for an edge dislocation three-fold symmetry about the dislocation is destroyed and four- or six-fold symmetry reduced to two-fold. The effect of such symmetry properties is most clearly seen in a scalar quantity such as dilatation. According to the anisotropic theory a screw dislocation along a three-fold axis has first order dilatation and since the symmetry is preserved a characteristic polar form is predicted (fig. 3).
,......-------_.. . . . 10-l1li:::'""-------_ /'
.... I
I
/
/
I
I
I
Fig. 3 The dilatation field of a [111J screw dislocation in Li (taken from Chou [21J). The unit of distance = b, and unit of dilatation = n/400.
For the stress field of an edge dislocation in the same two examples we obtain respectively
K1(xi (j12
).2 x
Db x
= 2n(xi- 2Axix~ +
1
1
).4 x
(70)
D'
where). 2 , A are further anisotropic parameters, and b cos 8 (cos 28 - b cos 8 cos 38) 4nS 11 (l - b 2 cos 38)r(1 -
(71)
£5)1/2
where tan 8 = x 2/Xl'
r2=
xi + x ~ .
In searching for the effects of the symmetry in particular solutions such as the displacement or stress fields it is necessary to bear in mind the difference between screw and edge dislocations. In the case of screws the Burgers vector lies along the dislocation itself and does not therefore affect the symmetry properties of this direc-
5. Applications ofanisotropic theory For many years the discussion of physical effects of dislocations has been largely confined to the approximation of isotropic elasticity. While this was hardly a satisfactory situation in view of the very few approximately isotropic materials it was made inevitable in practice by the mathematical complexity of the anisotropic treatment. The simplifications ofthe anisotropic theory which have now been achieved essentially overcome the earlier difficulties and it is now a relatively simple matter to use exact elastic expressions which take account of the effects of crystal properties. One area where rather precise knowledge of anisotropic strain fields has proved essential is that of transmission electron microscopy, the subject of chapter 18. In the case of mechanical properties the information available is of a less precise kind and the effects we look for are not so much quantitative as qualitative. It is interesting to glide, find that there are a number of alterations to be made in the simple laws climb, cross slip and pinning.
Ch.2
J. W. Steeds and J. R. Willis
160
5.1. Dislocation pinning One mechanism of pinning dislocations arises from their dilatational field~, which may be relieved by the atom size difference of an impurity by s~greg~tlOn of a component of an alloy. According to the isotropic th.eor.y there IS. no dilatation for a? undissociated screw dislocation and therefore nopmnmg of this type. The same. IS also true according to anisotropic theory when the disloc~tion. Bur~ers. vector lIes along a two-, four-, or six-fold axis. However, there is a fimte dilatation III all other cases, for example, in the case of the ia
0:
(S 11 + S 12)b3 b sin 38 zl = S15(l - b)1/22nr(1 - b cos" 38) An illustration of this dilatational field was given for Li in fig. 3. In referring to this figure we are evidently considering impurity pinning in greatly over-simplifi~d terms. In order to be a little more definite we may follow the steps of Cottrell and BIlby [22], neglect the effects of diffusion due to concentration gradients, ~nd examine the effect of hydrostatic pressure on a rigid impurity particle (we ought in fact to take account of the elastic properties of the impurity itself). It will lead us to an unexpected conclusion. In isotropic elasticity dilatational strain is directly proportional to hydrostatic pressure. This situation only persists in a few cases of high symmetry in practice, for example dislocations along [lOOJ or [1llJ in cubic crystals. In several other cases the line of zero hydrostatic pressure of an edge dislocation is the only real :oot of.a cubic equation and is not necessarily parallel to the slip plane. In rather amso~ropIc cases two additional zero lines appear in the hydrostatic pressure but not III the dilatation strain field. This situation is depicted in fig. 4 for a case where the first zero line lies in the slip plane (an example would be (001) [llOJ slip in a cubic crystal). We conclude that impurities can move perpendicular to equi-potential lines to l~cal positions of minimum pressure in regions of adverse volume change of the l~ttIce. In even more anisotropic materials two additional zero lines may occur III the dilatational strain field and one could then have the situation of impurities moving through a region of favourable volume change to rest in a region of adverse volume change. However, when we include diffusion, which will inevitably occur, the deepest pressure minima will be favoured.
of dilatational strain zeros of ...-rhydrostatic pressure ' "
Fig. 4 Positions of zero lines in the hydrostatic pressure and dilatational strain near a screw dislocation in a rather anisotropic medium. The ± signs refer only to the hydrostatic pressure.
§5.1
Dislocations in anisotropic media
161
Another new situation arises whenever the first zero line is not in the slip plane, for example the cubic {211 }ia
5.2. Glide and climb of edge dislocations The most common ways in which edge dislocations of the same type can reduce their energy is by forming polygonised walls (for edges of the same sign) or dislocation dipoles or multipoles (for edges of opposite sign). The lines of zero stress of the important o 12 field for glide interactions are commonly roots of cubic equations in the anisotropic treatment and will not generally take the orientations ±fn, tn. We need to examine the consequences of this behaviour. First we consider the effects of making the position of stable equilibrium for like edges at an angle ¢(in) to the slip plane, a case which occurs for example in cubic (211)ia
J. W. Steeds and J. R. Willis
162
Ch.2
A surprising hypothetical example can be envisaged where on account of extraordinary elastic properties the a 11 field acquires extra zeros while the a 12 field has two of its zeros at a relatively small angle to the slip plane (fig. 5). A freely gliding and climbing dislocation at the point A would then be attracted towards an identical dislocation at the origin. Since this seems implausible, it may well be that the configuration suggested here is incompatible with the crystal stability conditions. Note also that inthis case a jogged dislocation in pile-up would be particularly stable against climb on account of a force driving it back to the slip plane.
§5.4
Dislocations in anisotropic media
163
5.4. Dislocation interactions The influence ofelastic anisotropy on dislocation interactions may reverse conclusions based on isotropic elasticity. A case in point is the existence of dislocations with a [110J Burgers vector in b.c.c. crystals. Although energetically forbidden according to the isotropic theory it is possible in particular anisotropic cases and has been observed [25, 26J. Of course core energies should be included in the energy calculations but it has been found that there are a number of interactions which can be understood on the basis of elastic terms alone [27J. 5.5. Slip systems
011=0
011 =0
012=0
012=0
°11=0
---~~--- slip plene
Another area where it is important to include core terms in energy considerations is in the determination of a dislocation slip plane. In general the elastic contribution is not sufficient to decide the issue. One recent example where anisotropic theory gave a good account of observed behaviour is the case of «U. A phase transition at 44 K causes a marked change of elastic properties and calculations show that glide loops with a [1 OOJ Burgers vector have lower energy on (010) at low temperature and on (001) at high temperatures, in agreement with experiment [28].
Fig. 5 Position of lines of zero stress in the. a 11 and a 12 fields of an edge dislocation in a medium of extreme anisotropy.
6. Microscopicmeasurement of elastic properties
5.3. Glide and cross slip of screw dislocations
One idea of considerable potential which has yet to be explored is the measurement of the elastic properties of particular phases in a many phase material or of single crystals in a polycrystalline material. It has been established, largely by the work of Dr. Head's group in Melbourne, that quite accurate microscopic measurements can be made from dislocation images in transmission electron microscopy. The really interesting application of this work to phase stability in more complicated systems has yet to come. It seems appropriate therefore to outline some of the methods which might be used. Only relative magnitudes or anisotropic parameters can be determined with any accuracy.
The generalisation of properties of screw dislocations in anisotropic elasticity in some respects blurs the distinction from edge dislocations. Thus un-jogged pile-ups may be subject to cross slip forces and in rather anisotropic crystals screw polygonisation and dipole formation may occur. For cross slip we are concerned with the a 13 field. If this does not vanish in the slip plane as, for example, in the case of an f.c.c. crystal slipping on the (111)<1TO) system, screw pile-ups will be less stable than predicted by isotropic theory. On the other hand, if the material is so anisotropic that the a 13 field acquires two additional lines of zero stress then the pile-up would have a stable plane, resisting cross slip. Behaviour of this sort is conceivable in b.c.c. crystals although there is no known practical example. In order for stable configurations to be formed between gliding screw dislocations it is necessary to find a material which is sufficiently anisotropic that extra lines of zero stress appear in the a 23 field. Li is a practical example of this sort of behaviour and like screws are expected to form walls while unlike screws assume dipole configurations if gliding on distant slip planes. It may be however that rather than form polygonised walls the screws prefer to reorient, acquiring edge character, to form a lower energy configuration. The study of the plausibility of such a mechanism would probably require an analysis similar to that given in [15J for rhombus shaped loops.
6.1. Dislocation bends The instability of dislocations along particular crystallographic directions, referred to in sect. 4.2., leads to the formation of bent segments whose orientation may easily be determined. Although no general direct methods have been formulated for deducing elastic properties from measurements on the segments sufficient numerical data have now been accumulated to arrive at values by interpolation [29]. There may of course be a problem of uniqueness when this approach is adopted, and in some cases the equilibrium angle is rather insensitive to particular anisotropic parameters. 6.2. Equilibrium angle of dislocation dipoles The angle which an individual dislocation dipole makes with the slip plane is determined by the single dislocation stress fields. By measuring this angle it is therefore possible in principle to deduce values for anisotropic parameters. The method, which
J. W. Steeds and J. R. Willis
164
Ch.2
has been used by Forwood and Humble [30J, has the advantage that close dipoles are not much influenced by surfaces or other dislocations at distances considerably greater than the dipole separation. It suffers from the disadvantage that if the dislocations are dissociated it is necessary to know the stacking fault energy. 6.3. Image widths in the electron microscope One very simple approach which does not seem to have been explored relies on the variation of the displacement field with electron beam direction in the electron microscope. Consider the displacement field in eq. (68). For an incident beam along the x direction (tan () = yjx) -b
dU3
yjA 1/2 2 2n x + y2jA
dx
while for an incident electron beam along y dU3
= b
dy
AX
2n Ax 2
1 2 /
+
y2
The first result is equal to the isotropic case if we make the transformation A 1/2 X ---* x, and the second requires yjA 1/2 ---* Y i.e. we expand or contract the dimension perpendicular to the beam direction by A 1/2. Thus by rotating the foil containing a screw dislocation with this displacement field about the dislocation axis under identical diffraction conditions (same reflecting plane and deviation parameter) it is possible to deduce the value of A. This method has the attractive feature that unlike the two previous methods it will not be affected by a lattice friction stress.
References [1] J. Lothe, Phil. Mag. 15 (1967) 353. [2] L. M. Brown, Phil. Mag. 15 (1967) 363. [3] V. L. Indenbom and S. S. Orlov, Kristallografiya 12 (1967) 971 (in Russian); SOy. Phys. Cryst. (English Transl.) 12 (1968) 849. [4] V. L. Indenbom and S. S. Orlov, Zh. eksp. teor. Fiz., Pis'ma 6 (1967) 826 (in Russian); J.E.T.P. Letters (English Transl.) 6 (1967) 274. [5] J. R. Willis, Phil. Mag. 21 (1970) 931. [6] J. W. Steeds, Introduction to Anisotropic Elasticity Theory of Dislocations (Clarendon Press, Oxford). [7] V. Volterra, Ann. Ecol. Norm. Sup. 24 (1907) 401. [8] T. Mura, Phil. Mag. 8 (1963) 843. [9] J. R. Willis, Int. J. Engng. Sci.5 (1967) 171. [10] D. M. Barnett and L. A. Swanger, Phys. Stat. Sol. (b) 48 (1971) 419. [11] V. L. Indenbom and S. S. Orlov, Prikl. Mat. Mekh. 32 (1968) 414 (in Russian); J. Appl. Math. Mech. (English Transl.) 32 (1968) 414. [12] V. L. Indenbom and G. N. Dubnova, Fiz. Tverdogo Tela 9 (1967) 1171 (in Russian); SOY. Phys. Solid State (English Transl.) 9 (1967) 915. [13] S. S. Orlov and V. L. Indenbom, Kristallografiya 14 (1969) 780 (in Russian); SOY. Phys. Cryst. (English Transl.) 14 (1970) 675.
Dislocations in anisotropic media
165
[14] L. M. Brown, Can. J. Phys. 45 (1967) 893. [15] D. J. Bacon, R. Bullough and J. R. Willis, Phil. Mag. 22 (1970) 31. [16] S. G. Lekhnitskii, Theory of Elasticity of an Anisotropic Body, Trans. P. Fern (Holden Day, San Francisco, 1963). [17] J. D. Eshe1by, W. T. Read and W. Shockley, Acta Metall. 1 (1953) 251. [18] A. J. E. Foreman, Acta Meta1l. 3 (1955) 322. [19] A. K. Head, Phys. Stat. Sol. 6 (1964) 461. [20] J. W. Steeds, Phys. Stat. Sol. 3~ (1969) 601. [21] Y. T. Chou, Acta Metall. 13 (1965) 251. [22] A. H. Cottrell and B. A. Bilby, Phil. Mag. 42 (1951) 573. [23] F. R. N. Nabarro, Theory of Crystal Dislocations (Clarendon Press, Oxford, 1967). [24] J. W. Steeds, Phil. Mag. 16 (1967) 771. [25] L. K. France, C. S. Hartley and C. N. Reid, Metal Sci. J. 1 (1967) 65. [26] D. A. Smith, R. Morgan and B. Ralph, Phil. Mag. 18 (1968) 869. [27] T. Jessang, C. S. Hartley and J. P. Hirth, J. Appl. Phys. 36 (1965) 2400. [28] I. Saxl and J. Otruba, Czech J. Phys. B19 (1969) 459. [29] A. J. Morton and A. K. Head, Phys. Stat. Sol. 37 (1970) 317. [30] C. T. Forwood and P. Humble, Aust. J. Phys. 23 (1970) 697.
CHAPTER 3
Boundary Problems J. D. ESHELBY Department of the Theory of Materials University of Sheffield Sheffield, UK
© North-Holland Publishing Company, 1979
Dislocations in Solids Edited by F. R. N. Nabarro
Contents 1. Introduction 1. Introduction 169 2. Two-dimensional problems 171 2.1. Screw dislocations and circular boundaries 171 2.2. Screw dislocations in cylinders of general cross-section
2.3. Edge dislocations and circular boundaries 2.4. Other solutions for edge dislocations
195
3. Three-dimensional problems 198 3.1. Dislocations in a semi-infinite medium 198 3.2. Dislocations in plates and disks 204 3.3. Other three-dimensional solutions 208 3.4. Approximate methods 211 Addendum (1976) 216 References
219
183
177
The presence of free surfaces can sometimes affect the deformation of an internally stressed solid on an almost macroscopic scale: for instance, crystal whiskers containing an axial screw dislocation exhibit a twist which increases as their cross-section decreases. Such effects can be handled with the help of the theory of elasticity. On a microscopic scale relaxation due to the presence of the free surfaces of the foil affects the image contrast in thin-film electron microscopy. Of course, in this case we are really interested in the deformation of the lattice, but usually for want of anything better we have to treat the atoms as points embedded in an imaginary elastic continuum and deforming with it. Part of the free surface may take the form of an internal cavity. More generally, if the cavity is not empty but is filled with material having elastic properties different from those ofthe rest ofthe solid we have what may be called an elastic inhomogeneity. The interaction between dislocations and inhomogeneities is important in the theory of plastic deformation. These are some of the reasons why the subject-matter of this chapter has received the attention of solid-state theoreticians. The field has also served as a convenient exercise ground for applied mathematicians. We shall be mainly concerned with the following problem in linear isotropic elasticity: given the elastic field uf, p'(J due to some source of internal stress (a dislocation loop for example) in an infinite medium, find the elastic field of the same singularity in a finite body with a stress-free boundary S. The solution will take the form u'(' + u~m, prJ + p~j where the field u~m, p~j (which, by analogy with similar electrostatic problems, we may call the "image" field) introduces no new singularities inside S and is such that the surface traction p;jnj cancels the original traction prjn j on S. Evidently u;m, p;j is the field induced in the solid by known tractions -ptjn j applied to its surface. This is a standard problem in the theory of elasticity. The difficulties standing in the way of its solution can be made clear with the help of a theorem of Gebbia's [lJ which reduces the problem to one relating to an infinite solid. According to Gebbia the elastic field in a body whose outer boundary S is subjected to a traction T and a displacement U is the same as it would be if the material inside S formed part of an infinite medium provided S, now merely a surface marked out in the infinite medium, is covered with a layer of body force of surface density T and is the seat of a Somigliana dislocation whose variable discontinuity vector is equal to U. Figure 1 makes the theorem obvious. Insert the un deformed body into a perfectlyfitting cavity in an infinite matrix. Apply the surface traction T. Where the body pulls away from the matrix build it up with a thin layer of material until it meets the matrix, and where it would penetrate the matrix, scrape it away so as just to avoid interference. Finally weld body and matrix together over S. These operations do not affect the
170
J. D. Eshelby
Ch.3
elastic field within S and we are left with the situation envisaged in the theorem. The surface tractions have become built in as the layer of body force and the addition and removal of material over S generate the Somigliana dislocation. The latter can be regarded as being made up of a network of small dislocation loops each with a Burgers vector equal to the local value of U, and so the displacement inside S can be written down as an integral over S involving T, U and the elastic fields of a point force and an elementary dislocation loop acting in an infinite medium. (The field outside S is zero.) Expressed analytically the result is Somigliana's formula for finding the interior field in terms of surface traction and displacement ([2J p. 245). The trouble with this formula is that it requires us to know both T and U on the surface, whereas it is physically clear that a knowledge of T alone (which is all that we have) should be
Fig. 1 Gebbia's theorem.
sufficient for determining the interior field. In principle we could use Somigliana's formula to find the displacement at a particular point on the surface of S in terms of the known T and the unknown U at all points of S, solve the resulting integral equation for U and insert the answer into Somigliana's formula to determine the interior field. If the demand for the result is sufficiently urgent the above process could be carried out numerically; Cruse's method [3J seems to be well adapted to such calculations. Consideration of the interaction between a dislocation and an elastic inhomogeneity leads to a related problem. We begin as before with the solution uf in an infinite homogeneous medium, allow the elastic constants inside a surface S, which mayor may not embrace the singularities of ui", to change to uniform values different from those of the rest of the material, and ask for the resulting change in the elastic field. By an extension of Gebbia's result the solution may be made to depend on the determination of the field of a point force and of an elementary dislocation loop situated at a general point on the interface S. To see this, suppose that we have two replicas, I and 2, of the infinite medium, with the same singularities, but different elastic constants, symbolized collectively by c 1 and c 2' so that ui ( c 1) and ul c2) are the same functions of position with different values of the parameters c. Remove the interior of S in 1 and replace it with the corresponding region of 2, filling, scraping and welding as before. We are left with an inhomogeneous medium with the required singularities, but disturbed by the presence of a layer of body force of known density
§1
Boundary problems
171
[P;j(c 1 ) - P;j(c 2)Jn j and a Somigliana dislocation with a known misfit vector ur)(c 1 ) - uf(c 2 ) spread over S. The elastic field induced by the removal of these layers can in principle be found in the form of integrals involving the field of a point force and of an elementary loop at the interface. From the above it is clear that for a direct analytical solution of the free surface problem we should need to know the elastic field of a point force applied at a general point of the free boundary S. (There is a slight formal difficulty if S is a closed surface, e.g. a sphere, for there is no static solution corresponding to an uncompensated force. It may be turned by the device of introducing an auxiliary force and couple which do not affect the final result: see sect. 3.1.) Similarly, for the inhomogeneity problem we should have to know the field of a point force and of an elementary loop at the interface: actually the latter can be deduced from a knowledge of the effect of the force not only on, but also near, the interface. It is only in rather simple situations that these basic fields (Green's functions) can be found. Otherwise some special artifice has to be used for each class of problems. Early solutions of problems relating to the interaction of dislocations with boundaries were largely the work of physicists, who used a mixture of traditional and improvised methods. Later the subject was taken up by applied mathematicians. Many of the more recent results, analytical, numerical and mixed, cannot easily be treated briefly. We shall mention some of them, but mostly we shall illustrate the discussion only by some of the more presentable special solutions.
2. Two-dimensional problems 2.1. Screw dislocations and circular boundaries The simplest type of two-dimensional elastic field is a state of so-called anti-plane strain. Of the three Cartesian displacement components u, v, w only w is not zero, and it and the associated elastic quantities are all independent of the z-coordinate. Consequently the dilatation au/ax + av/ay + aw/az is zero and the displacement equilibrium equations reduce to Laplace's equation
r 2w =
O.
The non-zero stress components are
Pzx =
f1
aw/ax,
pzy =
f1
aw/ay,
where u is the shear modulus. The traction on a plane with normal (n x ' ny, 0) is parallel to the z-axis and has the value
Pzx nx + pzyny = f1 aw/an, where a/an denotes differentiation in the direction of the normal. T, =
The displacement associated with a screw dislocation of Burgers vector (0, 0, b) at the origin in an infinite isotropic medium is w = b(tan -1
y/x)/2rc
172
J. D. Eshelby
Ch.3
w = be/2n.
(1)
Since the expression (1) is independent of r, the traction, J1 ow/on = J1 ow/or, is zero on any circle r = const., so that eq. (1) as it stands already gives the solution for a dislocation with a hollow core of radius ao in a concentric cylinder of radius a and infinite length with a stress-free surface. The energy per unit length of the cylinder is
i
2 1r
o
fa ao
Boundary problems
173
0, ~2' ~1 respectively where ~1 and ~2 are related by
or in polar coordinates (r, e)
E = iJ1
§2.1
2
b a (grad W)2 r dr de = ~ In 4n ao
(2)
There are, of course, non-zero tractions on the cross-section of the cylinder. If a finite rod is cut from the infinite cylinder suitable forces must be applied to the ends if a state of anti-plane strain is to be maintained. If these forces are not supplied the rod develops a twist: see sect. 3.4. To treat the case where the dislocation is not at the centre of the circular crosssection it is convenient to have a few geometrical results from the theory of coaxial circles. We summarise them, together with some others which we shall need later. Figure 2a shows Cartesian coordinates (x, y) and three sets of polar coordinates (r, e), (r 2 , ( 2), (r l' ( 1 ) centred at the points 0, P 2' Pion the x-axis, with x-coordinates
~ 1~2
= a2 ,
{3)
so that .p 1 is the inverse of P 2 with respect to the circle r = a shown in the figure. The point P(x, y) may, of course, lie either inside or outside this circle. Figure 2b refers to the case when P lies on the circle. There are then several simple relations between the 8's and r's which we shall require here and later. In the triangles OPP 2 and OP 1P we have OP 2/OP = OP/OP 1 byeq. (3), and they have the angle at 0 in common. Hence they are similar and the relations
r.!«. = a/~l = 8 1 + 82 - 8 =
~2/a
on C 2,
(4)
n
on C z
(5)
easily follow. The changes b8 1 , b8 2 in el' 8z when P moves out radially by br to P can be found by projecting the segment PP' onto directions perpendicular to P P and P P. Thi . I~ gives -r z b8z = br sin P'PP 2 = br sin OPP z and r 1 b81 = br2 sin P'PP 1 1 = br sm OPzP, or b81 = b8z by eq. (4) and the relation sin OPPz/sin OPzP = ~z/a. Hence
o8t!or = o8 2/or
on C 2 .
(6)
Later we shall also need the relations (~1 - ~z)(rZ - a 2) = ~lr~ - ~2rf,
y
r~ -
ri
= (~1 - ~2)[(X - ~2) +
(7)
(x - ~1)]
(8)
which follow from
r;
=
(x - ~n)2
+ yZ,
n
='
1, 2.
Let Co in fig. 2a be a second circle for which P 2 and PI are inverse points. If its radius is ao and its centre is at (~o, 0) we have (~1 - ~o)(~z - ~o)
= a~
(9)
as the analogue of eq. (3), eq, (4) becomes
r.tr, p'
=
aO/(~l - ~o)
(a) Coaxial circles, (b) the point P lies on a circle.
(~2 - ~o)/ao
on Co,
(10)
while eqs. (5) and (6) hold also on Co. For a screw dislocation at P 2 in the cylinder r ::::; a with a stress-free surface we have only to put a negative image dislocation at Pl' The resulting displacement [4]
w = b(82
Fig. 2
=
-
( 1 )/ 2n
(11)
satisfies the condition J1 ow/on = J1 ow/or = 0 on r = a by virtue of eq. (6). Equation (11) can equally well be taken to refer to a negative dislocation at P 1 near . a circular hole. The change of viewpoint does not of course affect the fact that C is stress-free, but there is a Burgers vector b associated with the hole, and so eq. (11) actually represents a negative dislocation near a positive one with a large hollow core.
=
Ch.3
J. D. Eshelby
174
To get rid of the dislocation in the hole we may add the displacement w = b8/2n, which leaves the hole stress-free. Then, reversing the sign over all we get, using eq. (5)
w = b(81
(12)
82 + fJ)/2n
-
for a positive dislocation at P 1 near a circular hole. The dislocation in the cylinder experiences a force due to the image field represented by the second term in eq. (11):
(08 F; = bp~~ = -2; ay .
Ilb2
lIb 2
2; ~ 1
1 -
ub 2
E = 4"; In
P2
2
~ 2 = 2; a
~2 2
-
~~
-OE/O~2
2nw/b = a8
of a quantity -
~D
+
(13)
const.
which shows how the energy of the dislocation varies with position. If the dislocation has a small hollow core of radius ao we may tum the quantity in eq. (13) into an absolute energy by requiring it to agree with eq. (2) for ~2 = 0: E
=
E = ib pz dx = -Ilb2 4n
fa
(1 1) ---.- --
~o + ao X -
~2
X -
~1
dx.
This gives E = lIb In (a - ~2)(~1 4n (~1 - a)(~o 2
+
const.
(16)
+
f38 2
+
r < a
')I(n - fJ 1 ) ,
r
(17)
> 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
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
<
n,
')I'
= - 1, which
a
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
x',
y'l~x"y~y,
where x', y') = cp
+
+ ~ In /z - zll 2n
is the potential associated with the image field. This is easily checked if we bear in mind that blP(x', y')/2n varies twice as fast with x', y' as does cpim(x, y; x', y') with (x, y) held fixed near (x', y'). It is not hard to verify that lP satisfies the space-charge equation o2lP OX,2
+ o2lP = -4 e- 2 oy,2
+
(1)
+
+
(39)
([10J, [9J p. 203). Thus apart from positions of unstable equilibrium at maxima and saddle points the dislocation is everywhere attracted to the free surface C, near which E goes to negative infinity. This conclusion needs modification when the problem is treated as a three-dimensional one. These formulas can be given other physical interpretations. For example, cp may be taken to be the electrostatic potential inside a cylindrical condenser (transmission line) with C for outer and a thin wire of radius ao for inner conductor. E is the electrostatic energy. In other words the capacity per unit length is const./E. The force on the dislocation becomes the force on the wire. Or cp may be taken to be the stream function and w the velocity potential for a vortex line in a perfect incompressible fluid inside a tube C. E becomes the kinetic energy of the fluid. If the vortex is held fixed by a wire in its core the force on the wire is equal to the force on the dislocation. If the vortex is free the force on the dislocation becomes the drift velocity of the vortex in its own image flow, but turned through a right angle, so that the vortex describes a curve lP = const., that is, one of constant kinetic energy, as is to be expected. In the above the phrase "in suitable units" must, of course, be understood throughout.
+ a
+ b
Fig.4 A dislocation in a plate: (a) in the mid-plane, (b) not in the mid-plane.
either of the planes Y = ±idis straddled by a set of equally-spaced positive-negative pairs whose contributions to cp (or ow/oy) cancel on these planes so that they may be taken to be the stress-free surfaces of a plate with a dislocation at the origin. The argument of the logarithm in eq. (33) must have a zero at each positive dislocation and a pole at each negative one, and this will occur if we put fez)
= tanh (nz/2d) sinh 2% + i sin 2Y cosh 2X - cos 2Y
(40)
Ch.3
J. D. Eshelby
182
with
x
= tixfld,
Y = ny/2d.
It is easily verified that If(z)12 = 1 for z = x ± i!d, so that the expression, eq. (40), is the required mapping function. If the dislocation is at the origin, b 2n
tan 8 =
with
w =-8
sin (ny/d) , sinh (nx/d)
(41)
Pzy =
=
(42)
where we have used the relation sin 28 = 2 tan 8/(l this with the Peierls law [13J pz/x, y) = 2n; sm
b
+ tan? 8). We may compare
[w(x, y) - w(x, - y)J
na
(na)
2
Thus if we cut out the material between these planes and move them together until their spacing is a the elastic solution provides just the right stresses to support the Peierls force when it is switched on in the gap. If the dislocation is at (0, y'), eq. (33) gives cosh [nx/dJ - cos [n(y - y')/dJ
1
= - 2n lIn cosh [nx/dJ + cos [n(y + y')/dJ' b 2n
w = -tan
-1
({J
b 2n
= - tanh -1 (en 2Ax cn 2AY),
tan [n(y - y')/2dJ b -1 tan [n(y + y' + d)/2dJ --tan , tanh [nx/2dJ 2n tanh [nx/2dJ
y , = In [2d n ']•.
_ 1
(sd 2Ax ds 2AY),
where the elliptic functions of x have modulus k and those of y modulus k', We shall not give ({J and w for a general position of the dislocation, but only the function
+ dc 2 (2Ay', k')
- 1J
(cf. Greenhill [18J; our relation (39), evaluated at the origin, can be used to restore the additive constant which he rejects).
+ .. ,] .
r
b
(44)
with the usual notation [17J. If the dislocation is at the centre of the rectangle this gives
b
d tan - 1 - = +-a 1 [ 1 - -1 = +« 2d -2 3 2d
({J
fez) = sn AZ dn Az/cn AZ
w = - tan 2n
which must be satisfied on the planes y = ± ja across which the Peierls force is supposed to act. The elastic solution of eqs. (41) and (42) already satisfies the Peierls condition across the planes Y
A dislocation in a stress-free rectangle can be dealt with by repeated reflection of the array of fig. 4b in a pair of vertical lines [4J, by a suitable mapping [15J or by lifting the results from Greenhill's [16J treatment of a vortex line in a rectangular tube. The function which maps the rectangle with corners x = ±!a, y = ±!dinto the unit circle, centre to centre, is
K(k)/a = K'(k)/d = A
ub n ny. 4nw 2n 2d cot d sm T'
f1 b . 2n
183
where A and the modulus k of the elliptic functions are fixed by
and one component of stress is therefore 8w u 8y
Boundary problems
§2.2
(43)
The first term in w represents the array of positive dislocations shifted so that the dislocation within the plate is at (0, y') and the second the negative dislocations shifted by as much in the opposite direction (fig. 4b). Each of the planes y = ±!dis still straddled by compensating pairs. With x replaced by x(CSS/C4 4 ) 1/2 , w becomes the displacement in an 'anisotropic plate provided the dislocation axis and the normal to the plate are both two-fold axes of the crystal [14].
2.3. Edge dislocations and circular boundaries
In this section we try to do for edge dislocations what was done for screw dislocations in sect. 2.1. It is assumed that the material, infinite in the z-direction, is in a state of plane strain so that w, the z-component of the displacement, is zero and the other two components, u and v, are independent of z. (For a remark on the status of plane stress and generalized plane stress in dislocation theory see sect. 3.2.) We begin by recalling the theory of plane strain as developed in terms of the Airy stress function, and introduce some of the stress functions which will be needed. The details of a plane strain elastic field are all contained directly or implicitly in its Airy stress function X in terms of which the stresses are
Pzx = 0,
pzy = 0,
pzz = v(Pxx
+ p yy ) ;
(45)
the last relation comes from the requirement that ezz be zero (v is Poisson's ratio). The stress function is biharmonic, that is, it satisfies (46)
Ch.3
J. D. Eshelby
184
It is useful to remember that, if h is harmonic and rn is the distance from some fixed point, then h,
xh,
(47)
yh,
are all biharmonic. The displacement components, if required, can be calculated from X in the following way. They may be written in the form U
where
=
OX
OX
1 -, 2p, ox
((J -
-
1 2p, oy
v = l/J - - -
ox
eo
oP
oQ
oy
ox
= oy'
and F(z) = P
+
+
(49)
+
+
1 - v fZ il/J = - F(z') dz'
iy. Then (50)
ZQ
. 1- v ll/J = - - (z Zu
zo)F(zo)
1 - v fZ + -2/l
ZQ
,
(z - z)
(oP .oP) , -0 dz , -:jI -
uX
1
ds oy
0
-d (OX) - = 0 on C.
,
ds
ox
y'
q
+
Ix
+ my
(52)
ox/on
= 0 on C,
and the displacements are U
=-8 2n
b
+-
1
xy
-, 2
2n 2(1 - v) r
b V
(54)
1 - 2v
1
b
1
y2
= 2n 2(1 - v) In ;- + 2n 2(1 _ v) 7'
(57)
If the Burgers vector is (0, b) in the sense that u is single-valued while v contains a term +b8/2n we have
X = +Dx In r.
(58)
The simple displacement
v = 2n In r
(59)
has the harmonic stress function X
/lb
= - - (x8 + yin r).
(60)
n
It describes a compound defect made up of a dislocation (b, 0) and a concentrated force (0, - 2/lb). The x and y derivatives of eq. (60) give X
= const. 8
and
x=
const. In r,
(61)
representing, respectively, a concentrated couple and a centre (or rather line) of dilatation. From the difference of eqs. (55) and (60) we get 1 - 2v
(53)
around a curve whose projection onto the x-y plane is C. Since the addition of a linear function of x and y to X does not affect the stresses we can arrange that the plane represented by eq. (53) is of zero slope and height, so that the boundary conditions take the simplified form [4J X = 0,
(56)
b
The vanishing of these two expressions implies that the gradient of X is constant along C, so that if we plot X = x(x, y) as a surface over a horizontal x-y plane it is tangential to some plane
=
(55)
= /lb/2n(1 - v),
(51)
where the first term, a rigid-body displacement, may be dropped. Equations (48) and (51) combined are equivalent to the Cesaro line-integrals [19J for u, v in terms of strains and strain gradients. The traction on a curve C has x and y components -d(oX/oy)/ds and d(oX/ox)/ds where d/ds denotes differentiation along the arc of C. The condition that C be free of traction is thus
_~(OX) =
D
b
with arbitrary zoo If Q cannot be found from P by inspection insert a factor 1 = d(z' - z)/dz' in the integrand of eq. (50), integrate by parts and use the relation dF/dz = of/ox and eq. (49) to get
X
X = -Dy In r
with
iQ is an analytic function of the complex variable z = x
2/l
185
where Ojon denotes differentiation along the normal to C. We shall find it simplest to use eq. (54) wherever possible. However, if there are several separate stress-free boundaries we can use the condition of eq. (54) for one of them, but on the others we must require X to behave like eq. (53) on and near the boundary, with, in general, different constants q, I, m for each of them. The following are some stress-functions we shall encounter. For an edge dislocation with Burgers vector (b, 0) the stress function is
((J,
oP
tp
Boundary problems
(48)
l/J are a pair of conjugate harmonic functions. To find them from X write r 2 X = P and let Q be the harmonic function conjugate to P so that we have
qJ
§2.3
X
1
= 4n(1 _ v) y In r + 2n x8
(62)
for the unit force (0, 1). We begin by considering the case of an edge dislocation in a cylindrical rod with its lateral surface free of stress. The solutionsiwe shall obtain for this case will apply to a finite rod only if suitable forces are applied to.the ends so as to maintain the state of plane strain. However, we shall see in sect. 3.4. that the end-forces have no resultant
J. D. Eshelby
186
Ch.3
or resultant moment, so that their removal produces no drastic change in deformation, in contrast with what happens in the case of screw dislocations in rods. We also treat the complementary case of a dislocation in the neighbourhood of a circular stress-free hole, and then look at the more general problem of a circular elastic inhomogeneity. Volterra [20, 21J himself dealt with an edge dislocation in a tube; some of his results are reproduced by Love ([2J, section 156A). To find the Airy stress function for an edge dislocation interacting with free cylindrical surfaces concentric with the dislocation line we add to the basic
Boundary problems
§2.3
187
or for a » ao E=
ub? I n ( a-), 41[(1 - v) eao
e = 2.718 ....
(67)
If the dislocation in the tube has Burgers vector (0, b) the stress function is evidently the function in (65) with the factor - Dy replaced by + Dx. The stresses and displacements for this case are given by Love ([2J section 156A). We shall also need the stress function
r (ra a
2
X = - Dr sin
eIn r
X = In - - 1. 2
of eq. (55) multiples of the three other biharmonic functions r3
sin
e,
r
sin
e,
-: 1 sin
e
(63)
which have the same angular dependence. For a solid cylinder of radius a with the dislocation along its axis we need only the first two of them to satisfy eqs. (54): (64a) For a dislocation in an infinite medium with a hollow core of radius a we need only the last two: X
~
- Dy
[In ~ + i (;: - 1)J
or, rejecting terms which give no stress, simply
X = -Dy[ln r
+ !a 2/r2 ] ,
(64b)
which satisfies eq. (52) rather than eq. (54). For a dislocation in a cylindrical tube with inner and outer radii ao, a, we need all three of the functions (63). If we impose the condition of eq. (54) on, say, the inner surface we must require the x-surface to be tangential to the plane, eq. (53), at the outer surface, or, considering the symmetry, X = Ca sin 0, aX/or = C sin 0 on r = a. The result is X
+ r2 ( 2 a + a~
= - Dy [ In -r - -I ao
a
2
2
I
;g)]
ja Pxix , 0) dx ao
-
1)
(68)
obtained by striking out - Dy in eq. (64a). It is made up of a two-dimensional hydrostatic pressure and a centre of dilatation, eq. (65), combined so as to make the circle r = a stress-free. For r > a it represents the field in an infinite solid subjected to a uniform pressure perturbed by a circular hole. The centre of dilatation takes account of the inward movement of the matrix when a plug is removed from the compressed material to leave the hole. Suppose next that the edge dislocation is excentric, say at Pi~2' 0) in fig. 2a. To find the stress function for a dislocation in a cylinder, circular or not, with stress-free boundary C we have to subtract from the stress function for an infinite medium another biharmonic function which has the same value and normal derivative on C, so satisfying eqs. (54), but which is free of singularities inside C. If C is a circle there is in fact a formula, similar to Poisson's integral formula for a harmonic function, which determines interior values of a biharmonic function from the boundary value and normal derivative [23J, but we shall not make use of it. We start with the case where the Burgers vector (b, 0) is parallel to the x-axis, Following our treatment of the screw dislocation we begin by adding together the stress functions -Dy In '2' +Dy In'1 for a real dislocation at P 2 and an equal and opposite image one at P1 to form the first tentative solution
Xo = -Dy In t
(69)
with (70)
(65)
Photoelastic pictures of the associated stress field have been given by Corbino [22J. The elastic energy for the last case can be found as the work required to establish the dislocation by sliding the two surfaces y = ± 0, ao < x < a through a distance b:
E = !b
2
= !b[oX/oyJ: o
The extra factor a/~2 = ~ .!« which we have inserted in t ensures that t is unity in the circle r = a, so that Xo satisfies the first of the boundary conditions (54), but not the second. Near t = 1, In t can be expanded in powers of t 2 - 1: In t =
! In t 2 = ! In [1 + (t 2 -
1)J =
!(t 2 -
1)
+ ....
The first term of the expansion reproduces the value and gradient of In t correctly on the circle. Hence if we subtract it from In t we get a function (66)
X(t) = In t - !(t2 - 1)
Ch.3
§2.3
which satisfies both of the conditions of eq. (54) on the circle. The same will be true of the product of X(t) and a function of x and y. Hence
and
188
J. D. Eshelby
x=
Boundary problems
solves the problem, provided it is biharmonic. Fortunately it is, having the form of the fourth of the quantities (47), with h = y /ri . Dropping a term const. y, which gives no stress, X becomes [24-26]
r2 1 ~ 1 r~ X= -Dyln-+-Dy--· r, 2 ~2 d
(77)
for a dislocation with Burgers vector (b, 0) at (-~, 0) in the semi-infinite solid x < O. When the Burgers vector is (0, b) instead of (b, 0) the calculation is more troublesome. Suppressing the factor D we start with the simple object-image stress function (x ~2) In r 2 - (x - ~1) In r 1 (cf. eq. (58)) and simplify it to
Xo = (x - ~2) In t
(72)
Or again, replacement of r~ in the second term by, say, r~ ri likewise merely adds a term const. y and so, byeq. (8), eq. (72) is also equivalent to
2~
1
E = 'ibDlneao
(71)
-DyX(t)
189
(78)
by adding a harmless centre of dilatation, eq. (61), at the image point. From the previous argument it is clear that (x - ~2)X(t) satisfies the boundary conditions and has the right singularity, but unfortunately it is not biharmonic, since (x - ~2)/d is not harmonic. We therefore write
(73)
X = Xo - Xl and try to choose X1 so that it is biharmonic and singularity-free inside C and has a value and derivative which coincide with those of Xo on C. We look for Xl in the form
with
The stress functions, eqs, (72) and (73), satisfy the general boundary condition eq. (52) rather than the special form eq. (54). The image stress function is made up of the simple image term Dy In r 1 together with the last term in eq. (73), which can be written in the form
Xl = 9
E = tbD In
a
2
-
e
eaa o
2,
(75)
where the additive constant has been chosen so as to agree with eq. (67) for ~2 = o. In the limit oflarge a, ~ 1 - ~2 and x - ~ become 2~ and x in the notation of fig. 3, and we get (76)
a 2 )h
-
(79)
with harmonic 9 and h. (Equation (79) can actually generate any biharmonic function, but we make no use ofthis fact.) As Xo is already zero on C, 9 is zero everywhere. Since a/an is equivalent to a/or on C we have I 0Xl a an
(74) representing a dislocation dipole and a doublet of centres of dilatation (eq. (61)) supplementing the image dislocation at Pl. The image stress is derived from eq. (72) with the term - Dy In r2 omitted. The non-logarithmic term contributes nothing to the image stress at P 2' though it does so elsewhere on the x-axis. We may say that the image force is the same as that due to a simple negative image dislocation at PI represented by a term Dy In r l' the additional singularities separated out in (74) having no effect. The image force is thus 1/(1 - v) times what it is for a screw dislocation in the same situation, eq. (14), and so the energy is
+ t(r 2
= h on C 2
(80)
and
OXo = (X _;;) :::l ~2 un
[(x - ~ 2
2
r2
1)
(y riy) YJ-
x -- ~ -X + - - -
ri
r
r~
r·
on C 2
or (81)
where we have used the facts that r = a and that ri~2 = d~l on C 2, by eq. (4). Before eq. (81) can be compared with eq. (80) it must be cleaned up with the help of the various identities which hold on C 2. First we must use eq. (4) to replace r-;2 by (~d~2)r12; otherwise there would be an unwanted extra singularity at r2 = O. Then we must eliminate (x - ~2) in favour of (x - ~1)' since (x - ~l)/d is harmonic, but (x - ~2)/d is not. This can be done with the help of the relation in (8), which holds generally. This gives
~ OXo
a
an
=
-(~1
-
~2) (x
- ;1 + -;) a
~2rl
on C 2 .
(82)
190
Ch.3
J. D. Eshelby
If we now require h, eq. (80), to agree with the right-hand side of eq. (82) not only on C but throughout its interior, the Xl of eq. (79) satisfies the required conditions. Hence finally we have
X = D(x - ~2) In t
1 2 + lD(r - a 2 )(~1 -
~2)
{x-;:-2-~ + 1
<"'2r1
21} .
(83)
a
§2.3
Boundary problems
The elastic energy corresponding to eq. (87) is evidently eq. (75) with 1 and 2 interchanged. When the field (64b) is added there is an extra term bDR 1
= D(x -
X
~2)
In -
r1
+
D(~l -
~2)(X
-
x - ~1
- ~) --2r1
+
r lD(~l - ~2) 2 a 1
= ~1
Db -
~2
Db -
2 ;:2 + ;:2) E = lbD In a ( eaa., a
(84) E = ibD [In
(90)
~i ~ a' - ~;J + const.
(91)
The additive constant cannot be given a definite value, since in an infinite matrix the energy is formally infinitive even if we cut out a stress-free hole around Pl' Similarly, if the Burgers vector in the situation just considered is changed from (b, 0) to (0, b) we start by interchanging 1 and 2 in eq. (84) and adding a term (92)
':> 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 <
o.
I
The constant C' must stay unchanged to give the correct singularity at PI and the boundary conditions fix the other two. The second term in eq. (105) can be modified similarly, and then V" may be treated in the same way. In this way we end up with two known combinations, eqs. (104), of T and P from which Pxx and Pvv can be extracted, and hence also Pxv by integration of oPxvloY = -oPxxlox. The 'numerical coefficients in the result depend on the elastic constants and on the roots a', a". Head shows that if the two Poisson's ratios are the same only a' + a" and a' o" are involved, so that the coefficients are free of radicals. Comparison with Dundurs' results [for example eq. (96) aboveJ suggests that the same is true in the general case. Indeed this must be so since the T and P found by solving the simultaneous equations (104) are rational symmetric functions of a' and a" [32J though it is difficult to make out the details. Dislocations near plane interfaces in anisotropic media have been treated by Pastur et al. [33J, Head [34J, Gemperlova [35J and Tucker [36].
of the first pair and
o [( f3 + -1 ) T + (1-- 2v - -,f3P I -ox 2f.1 4f.1 2
)J
(103)
of the second, and requires a, f3 to be chosen so that (103) is a multiple of the x-derivative of (102). If f3 is fixed this can be done by varying a, but a would be different in the two media. However if both a and f3 are allowed to vary they may be chosen to be the same in the two media. This leads to a quadratic equation for a, with roots ::I.', ::I." say. The corresponding quantities (102), say V'
= A'T + B'P,
V"
= A"T + B"P
(104)
are then harmonic, continuous across x = 0 and, because of (103), the quantities K' oV'/ox and K" 0 V"jox are also continuous across x = 0 where K', K" depend on ::I.', x" and the elastic constants. These are the boundary conditions for the potential at the interface between two dielectrics, so that we expect that point singularities can be dealt with by imaging. For a dislocation with, say, Burgers vector (b, 0) at PI (fig. 3) in a homogeneous medium we have (105)
2.4. Other solutions for edge dislocations There has not been much analytical progress in extending the results of the last two sections to non-circular boundaries, because of difficulties in handling biharmonic boundary value problems. Since the same difficulties also apply to other physical phenomena governed by the same equations there is not, as there was in the case of anti-plane strain, an extensive store-house of results to draw on. Equations (46) and (54) also govern the transverse deflection of a plate clamped around C. If the plate is deflected by a point force at ~ the displacement is 2 - wI(r), W = const. (r - ~)2 In
Ir - '1
where on C WI has the same value and normal derivative as the first, singular, term. When regarded as a stress function W represents a wedge dislocation (disclination) made by cutting out a narrow wedge of material with its apex at ~ and closing and welding together the faces of the gap. If w is differentiated with respect to ~x or ~y the resulting stress function has a singularity of the form const. (x - ~x) In ~I or ~I and so represents an edge dislocation [4, 37]. A number const. (y - ~y) In of solutions have been given for clamped plates deflected by a point force at a single point of high symmetry. They can be used as the solution for a disclination but, as they cannot be appropriately differentiated they are of no help in the dislocation problems. Indeed the only useful solution seems to be Michell's for a force at an
Ir -
Ir -
J. D. Eshelby
196
Ch.3
arbitrary point in a clamped circular plate [2]. It can be used to reproduce some of the formulas of the last section. Seeger [26J, quoting from the work of Leibfried and Dietze, gives the value
E =
JJh2 4n(1 - v)
y In ( -2dc o sn -' ) na o
d
(106)
=
x I - v
---
(107)
with the qJ of eqs. (43). The forces between the dislocations increase by a factor 1/(1 - v) and so the potential energy of the real dislocation in the field of the images is 1/(1 - v) times the energy eq. (37) with the ifJ of eqs. (43), and this agrees with eq. (106). However, eq. (107) only satisfies the first of the boundary conditions of eqs. (54), and the normal but not the tangential traction is zero at the faces of the plate. To liquidate the tractions completely we should have to carry out the usual procedure in which by repeated imaging the boundary conditions are alternately satisfied on one of the planes at the expense of a certain, but ultimately vanishing, violation of them on the other. At each imaging an image dislocation acquires a family of singularities of the type of eq. (74), but in addition its previous family acquires a family of its own. Hence finally we have an array of image dislocations each accompanied by a group of singularities which increases in complexity as we move outwards from the real dislocation in the plate. The implication of eq. (106) is that these extra singularities, either group by group or collectively, exert no force on the real dislocation. Equation (106) is perhaps subject to the limitation already noticed in connection with eq. (75); see the discussion following eq. (95). Lee and Dundurs' [38J comparison of eq. (106) with their own numerical results also suggests this. The energy of a dislocation near a crack, an infinitely flat elliptical hole, is of interest in fracture mechanics and can easily be found with the help of the formalism developed in that subject. Suppose a straight crack has its ends at (0, 0) and (l', 0). First calculate the three stress intensity factors at the tip (I', 0) in the form of suitably weighted integrals of the unperturbed dislocation stresses along the proposed site of the crack, and from them find the energy release rate G(I'). (See, for example, ref. [39].) By definition G(I') d/' is the reduction in total energy when the tip extends by dl'. (The total energy is the sum of the elastic energy and, though it does not concern us here, the potential energy of the loading mechanism if any.) Thus if the crack
197
establishes itself by growing from zero to a finite length I, the end (0, 0) staying fixed, the energy of the dislocation falls from Eo to E
for the energy of an edge dislocation, with arbitrarily oriented Burgers vector, distant y' from the mid-plane of a plate of thickness d. This result is remarkably simple when one recalls the difficulty of solving elastic problems relating to slabs, but the promised details have not been published. If in fig. 4b the array of screw dislocations becomes an array of edge dislocations with Burgers vectors (0, ± b) the appropriate Airy stress function is easily seen to be X
Boundary problems
§2.4
~ Eo -
f:
G(l') dl'.
The interaction energy between a dislocation and an inclusion of any shape can be found very easily with the help of a simple general result [40J provided the difference in elastic constants is small. Suppose that in an initially homogeneous body the Lame constants )1, A change to )1 + b)1, A + bA where b)1 and bA may possibly vary continuously from point to point in addition to being discontinuous across interfaces. Then an estimate for the increase in elastic energy is the integral (108) taken over the region where b)1 and bA are not zero, using the original strains eij' The error in eq. (108) is of order (b)1)2 even though the strains are altered by quantities of order b)1. (For brevity we shall suppose that b)1 and bA are of the same order.) In eq. (108) it is assumed that the strengths of the sources of internal stress (or rather strain) are unaffected by the change in the elastic constants; in the present case this means that the Burgers vectors of the dislocations are not altered. If the stresses are partly produced by external loads eq. (108) gives the change of the total energy, made up of the elastic energy and the potential-energy of the loading mechanism. There is a similar result for anisotropy. To verify these statements suppose that despite the change in elastic constants the original e i j remain unchanged. The change of energy is then exactly the quantity of eq. (108), even for finite bJ1, bA. To maintain these "wrong" strains it is necessary to impose a certain volume distribution of body force and also suitable layers of force on the interfaces across which the elastic constants change abruptly. They could be found by substituting the old strains into the new equilibrium equations with body force and the new interface boundary conditions, but we know in any case that they are of order bJ1. If they are now relaxed the change of displacement is of order b)1 and an amount of work £1 of order (b)1)2 is extracted from the system, and so eq. (108) is still correct apart from being too large by an amount E', of order (b)1)2. Equation (l08) is an example of a general class of theorems of which the Hellman-Feynman theorem [41J in quantum mechanics is the best known. The correction - E 1 which makes eq. (108) correct apart from an error of order (b)1)3 can also be calculated without solving a boundary value problem since with sufficient accuracy the displacement induced by the restraining body forces can be calculated using the elastic Green's function for the original uniform medium, eq. (112) below. Equation (108) can give not too bad results even if b)1, bA are not particularly small. For example it reproduces the functional form of the two terms in eq. (97) correctly, and with, say, VI = V2 = :L J11 = 1.25 J12 it gives 0.10 and 0.03 for their coefficients ~(P + Q) and ¥3P - Q) in place of the correct 0.11 and 0.04. If the angle between
Ch.3
J. D. Eshelby
198
the Burgers vector and the x-axis is 8 rather than zero eq. (97) becomes 2 ~2 _ a a2 a2) E = ibD 1 ( L In 1 ;d + M ;:2 cos 28 + N ;:2 ~i
~i
(109)
~i
where for small bfl, Land M are of order bfl and N is of order (bfl)2. Equation (108) reproduces the terms in Land M but, naturally it fails to give the term in N, which, however, appears in the correction - E 1 . Reference [7J contains an extensive list of papers on two-dimensional inhomogeneity problems, to which may be added the following. The problem of an edge dislocation in a semi-infinite solid with a finite plane inhomogeneous surface layer is treated in ref. [42J for isotropic and in ref. [43J for anisotropic media. Reference [38] deals with the case when the dislocation is in the surface layer.
3. Three-dimensional problems
found the field of an elementary dislocation loop in an infinite solid. (In his equations (7.18) and (7.20) T is a misprint for w.) He actually used a Fourier method to annul the normal tractions (cf. also Maruyama [45]). Alternatively we could annul the normal force and double the tangential forces by subtracting u?, and complete the calculation with the help of the expressions for a tangential surface force. Though this is less convenient the final result may look simpler when written in terms of uf - u? rather than u f + u? (see below). The solution for an infinitesimal loop or other singularity in a semi-infinite medium may also be made to depend upon known expressions for the elastic field of a point force in a semi-infinite medium as given in refs. [46-50] and in two dimensions in [51-53]. The field of an infinitesimal loop at the origin in an infinite medium with Burgers vector b i , normal n i and area dS can be written in the form [54J OCJ
dU i = flbjnk dS
3.1. Dislocations in a semi-infinite medium
199
Boundary problems
§3.1
(a c. aX k
+
a v; aX}
2v
aVim)
+ 1 _ 2v bjk aXm
(111)
'
where
A problem of importance in, for example, electron microscopy and geophysics is the determination of the effect of a free surface on the elastic field of a dislocation loop. In the geophysical applications it will usually be enough to consider a semi-infinite solid (half-space) and this case may also be adequate in the electron microscopy application, or at least it represents the first step towards the solution for a dislocation in a plate. In dealing with any elastic problem concerning a half-space it is useful to introduce what may be called a geometrical image field. A physical picture of an original displacement uC: may be produced by drawing an arrow at every point of space to represent the displacement there. Reflect the whole pattern of arrows in the horizontal plane X 3 = 0 and interpret the reflected pattern as a picture of the geometrical image displacement field u?We evidently have
i = 1,2 and (110)
A few sketches combined with eq. (110) will show that the original and image loops are identical for horizontal or vertical prismatic loops and for vertical shear loops with a horizontal Burgers vector, and equal and opposite for horizontal shear loops and vertical shear loops with a vertical Burgers vector. From eq. (110) we easily get
Hence if we add the G-field to the co-field the shear force on X 3 is annulled and the normal force is doubled. The effect of removing the normal forces can then be calculated, say from Boussinesq's expression ([2J p. 192) for the field due to an arbitrary normal load applied to the surface of a semi-infinite solid. In this way Steketee [44]
(112) is the i-component of the displacement produced by a unit concentrated force at the origin of an infinite medium, directed parallel to the xraxis ([2] p. 183), or explicitly duC: = -bjn k dS[(1 2v)(bij lk + biklj bjklJ + 3lJjlk]/8n(1 - v)r 2 with li = x fr. If the loop is at position
~i
eq. (111) becomes
a
dzz, = - pb n, dS [ bj1a~k
+
a bkl a~j
2v
a]
+ 1 _ 2v bjk a~l
Vil(x s , ~s)'
(113)
where Vil(x s , ~s) denotes the displacement of eq. (111) translated by ~i' We have taken advantage of the fact that it depends only on Xs - ~s to replace a/ax; by - a/a~i' and to save later re-writing the affix 00 has been dropped. Since ~i denotes the point of application of the point force, eq. (113) states that dU i is the same as the displacement field due to a certain distribution of force doublets at ~i' with zero resultant and, since the expression (111) is symmetric injk, no resultant moment. Let us next give a new meaning to Vil(x s , ~s) in eq. (113) and allow it to stand for the displacement due to a point force at ~i in a semi-infinite medium with a stress-free surface. (It is of course no longer a function of X s - ~s only.) The new Vii is made up of the old one plus an image field with no new singularities, and so eq. (113) now gives the field of the loop in a semi-infinite medium. More generally we may take Vu(x s ' ~J to refer to a point force in any body with a partly stress-free surface provided it extends to infinity in some direction, so that it is legitimate to think of an apparently uncompensated point force acting on it, though in fact it is balanced by forces of order r -2 at infinity. Even for a closed surface, say a sphere, where this is not allowable, we can save eq. (113) by re-defining Vil(x s , ~s)' Let it stand for the i-component of the displacement produced
Ch.3
J. D. Eshelby
200
by a point force parallel to the x-axis at ~ 1 in a finite body with a stress-free surface, together with an equal and opposite force at an arbitrary fixed point P (independent of ~J and a point couple at P whose moment balances that of the two forces. Then eq. (113) gives the field of the elementary loop alone since the resultants of the forces and couples at P associated with the force-cluster at ~i are zero. Using Mindlin's results and eq. (113) Bacon and Groves [54, 55] have derived the displacement field of an elementary loop in a semi-infinite medium and have presented their results in a remarkably compact form. The loop is at (0, 0, e) in the semi-infinite solid 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,
°
202
J. D. Eshelby
Ch.3
where (j) = Clr. This equation gives a solution of the elastic equations with any harmonic ip, If, further, (j) depends on rand z but not on e we have Ue
Pez = J.1(ouel oz),
= o(j)/or,
Per = J.1((ou e/or) - ue/r)
§3.l
+
z)
oZ(j) 2J.1ui = x 3 - ax, oX3
(117)
+
zZ
=
(118)
203
a purely normal traction on x 3 = 0, since, as we have seen, each elementary objectimage loop pair does so. Yoffe cancels this normal traction by adding the elastic field
with the remaining components zero. If we put tp = cir, the potential of a point charge.ps, is proportional to r- 3 on z = 0 and so does not cancel the stress in eq. (116) on z = O. If we integrate clr along the negative z-axis it becomes (j) = c In (R
Boundary problems
o(j) ax,
+ (1 - 2v)-,
i
1,2
=
(121) with harmonic (j). The stresses which contribute to the traction on x 3 = 0 are
with RZ = r Z
X
Z
+ yZ +
zZ.
This is the potential of a uniformly charged wire along the negative z-axis. With eq. (118), eq. (117) gives a stress proportional to r- z on z = O. We therefore integrate once more to get (j) = c[z In (R
+
z) - RJ,
(119)
the potential of a semi-infinite wire whose charge is proportional to 14 With eq. (119), eq.(l17)givesPez = cuir cnv z = owhich cancels thePez ofeq. (116)withc = -b/2n, as required, and the image field is . = e
U 1ffi
b r 2n R + z
J.1b r p i ffi _ ez - - 2n R(R + z)'
--~~,
p:;' =~(;: + R
~
J
and 03(j) 02(j) P33 = X3 -3 - - 2 ' OX3 OX 3
so that the surface will be stress-free if 02(j)/OX~ is chosen to be equal in value to P33 for the angular dislocation on x 3 = O. We refer to the original paper for the general expressions for (j) and the field of the angular dislocation and only consider the case where the dislocation meets the surface normally. The angular dislocation then becomes a straight dislocation along the x-axis, with the elastic field of eq. (57), supposing that its Burgers vector is parallel to the x-axis. The appropriate (j) is then
(120)
Note that l/(R + z) may also be written in the form R/r z - zlr". The complete field is the sum of the expressions (116) and (120). With the help of her theory of angular dislocations Yoffe [60J has given an elegant treatment of the general case where a straight dislocation, of arbitrary Burgers vector, meets the surface of a semi-infinite solid at any angle. Figure 5a shows a dislocation meeting the free surface of the semi-infinite solid X 3 < O. We cannot allow it to terminate in the infinite solid with which we have to start the calculation, and so we let it continue along the negative x 2-axis so as to form an angular dislocation. (In physical terms the dislocation produces a growth step on the free surface, say along the negative xz-axis. When a second semi-infinite solid is welded onto the first the step gets built in as a dislocation.) The angular dislocation can be analysed into a network of infinitesimal loops, one of which, together with its geometrical image, as defined by eq. (110), is shown in the figure. We suppose for the moment that the Burgers vector is horizontal. Then the argument following eq. (110) shows that if the original and image elementary loops are described in the same sense their Burgers vectors are the same. Thus if an image dislocation is synthesized from the image loops its horizontal arm will cancel that of the original dislocation, and the remaining parts of original and image may be joined up to form a single angular dislocation whose field is known (Yoffe [61J). It produces
vub [ X Zx 3 (j) = 2n(1 - v) x 3 In (R - x 3) - R - x
]
3
with R2 =
xi + x~ + x~.
The total shear stress PIZ in X 3 = 0 has the same functional form as for the infinite dislocation, but multiplied by a factor 1 + v(1 - 2v), or say about 1.1. Two like
o a
D
b
Fig. 5 (a) An angular dislocation, an elementary loop, and their images in a free surface. (b) An acuteangled and an obtuse-angled dislocation combine to form a straight dislocation and its image.
J. D. Eshelby
204
Ch.3
parallel dislocations will thus experience a more than normal repulsion near the surface and should splay apart there. Actually the contrary behaviour is sometimes found in ionic crystals; evidently elastic image effects will not explain it [62]. It is possibly due to an electrostatic surface effect [63J. If next the Burgers vector of the dislocation is vertical the Burgers vectors of the elementary loops in fig. 5a are opposite, the horizontal arms of the original and image dislocations do not cancel but reinforce one another and their oblique arms cannot be joined up to form a single angular dislocation. The configuration still produces, by our general argument, a purely normal traction on x 3 = 0, but the function qJ required to annul it would, so to speak, have to waste a great deal of effort in cancelling the singularity along the negative x 3-axis. To avoid this we swing the horizontal arms round until they lie along the positive x 3-axis to give a pair of angular dislocations (fig. 5b) one acute, the other obtuse. Yoffe finds that they produce the same shear stresses on X 3 = 0 as would an infinite screw dislocation along the x-axis, and these can be cancelled by a field like that ofeq. (120) but reflected in X 3 = 0, since the solid now occupies the half-space X 3 < O. They also give a normal stress which can be cancelled by the field of eq. (121) with a suitable qJ. We again refer to the original paper for the details.
§3.2
Boundary problems
which falls off with increasing z and agrees with eq. (124) for z = O. From eq. (117) the corresponding displacement is
im = - -b f.oo e -kzJ
Uo
e±ikz11(kr),
2n
ub 1 PzO = -PzO = - 2n -;"
~
-
~
r
J,(kr) dk
cosh kd
0
J (kr dk. 1
)
(126)
k
00
L
sech kd = 2
(_l)n exp [ -(2n
+
l)kd]
n=O
converts eq, (126) into a sum of simple image terms like eq. (120); for each of them the origin of z is displaced and that part of the z-axis which is the seat of couples lies outside the plate: b
=-
00
2n
x
L (- l )" n= 0
L,
+ z + [(d,'+ 2)2 + ,2J'/2 - d, _ Z + [(d,'- 2)2 + r 2J1/2}
with
d; = (2n + l)d.
(123) U
ub Poz = nd
b z o= - 2 nr
L (sin. 21 nn )K
1
b . 1 nn) -2 + -L(sm 2 tt
ntt
nitz K 1 (nnr). sm-, 2d 2d
(nnr) nttz 2d cos Ti'
(124) pb2 z - ubdi...J(sm '" . 1 nn)K (nnr) . nttz PrO -_ 2 z -2d sm-, tt r tt 2d
which may also be written as P:':l
k
The expansion
where k is arbitrary and J 1 , Y 1 , 11' K 1 are the Bessel functions usually so denoted. On a free surface z = const. we must have 00
dk ( k r)-, 1
Alternatively eq. (126) can be expressed as a series of modified Bessel functions by the theory of residues. The complete field, original plus image, is
which has the basic solutions
im
0
U~m = _ ~ f.oo sinh kz
u ~m
(122)
2n
which by a standard result reproduces the displacement of eq. (120). Similarly, for a plate with stress-free surfaces z = .xd we insert a factor cos kz] cos kd into the integrand of eq. (125). The corresponding displacement is now
3.2. Dislocations in plates and disks The elastic field for a screw dislocation traversing a plate may be found by an extension of the analysis for the half-space [59]. We assume that the image field is of the same form as eq. (117). Then both u~m and P~o satisfy the single equilibrium equation
205
u, = 0,
Prr = POO = pzz = Prz = 0
(127) (128)
(125)
by using the standard results J 1 = -J~, Jo(O) = 1, J 1(X) = o. To extend the expression in eq. (125) into the interior of the semi-infinite solid z > 0 we insert a factor e- kz into the integrand, so making it a solution of eq. (122)
with summation over integral n (terms with even n are actually zero). The solution of eq. (127) can be verified without reference to the method by which it was found. By eq. (123) each term of Uo satisfies eq. (122), and in Pzo each term vanishes on z = ±d. Since Kn(x) falls off rapidly as x increases the Bessel function
206
Ch.3
J. D. Eshelby
terms are unimportant a few multiples of d away from the dislocation axis, and the field components reduce to U
e
b z 2n r
= - --,
Pez = 0,
P
}lbz
re
= - -2,
n r
(129)
together with eq. (128). This simple field in fact satisfies the equilibrium equations and the boundary conditions on z = ±d and there seems to be no need for the Bessel function terms. However, the field of eq. (129) is evidently the average of that of eq. (120) and of the same field reflected in z = 0 but with couples of opposite hand. There is thus a distribution of couples along the z-axis not only in the image but also in the plate itself. It is the task of the Bessel function terms to cancel the part of the couple distribution inside the plate without upsetting the boundary conditions at the surfaces. That they do get rid of the extra axial singularity may be verified by expanding the term -bz/2nr of the complete expression for ue (eq. (127)), asa Fourier series in sin (nnz/2d), and combining each term with a corresponding Bessel function term. The result is the original series for U e with the term - bz/2nr deleted and each K 1 (x) replaced by K 1 (x) - x -1, which is non-singular at x = O. A screw dislocation in a plate exerts a total force F
= b'
d
J
Pez dz =
-d
4bb' n
-2
I -n1 K (nnr) 2d 1
Boundary problems
207
The displacements and stresses of eq. (127) can be adapted so as to give.the elastic field in a circular disk of outer radius a and inner radius a o, the latter representing the hollow core of the dislocation. Consider the Bessel-function terms for one particular value of n taken by themselves. They satisfy the governing equations and the condition that the plate surfaces be stress-free. Byeq. (123) this will still be true if we replace K 1 in U e by a linear combination AnI1 + BnK 1 of itself and the second solution, 11 (x), of the equation satisfied by K 1 , with corresponding changes in the other terms. Ifwe treat the terms for all n in this way the boundary conditions at the surfaces of the plate are not upset, and we can choose the An' B; so that the cylindrical surfaces r = a, r = a., are also stress-free. The final result is U
z
b =-8
2n'
p.; = Pee = pzz = Prz = 0, (130)
n odd
on another screw dislocation with Burgers vector b' distant r from it. If r is much less than d the force has a value bb' 2nr
§3.2
where An = 4b2 ~3 sin Inn n n 2 x
F= -·2d
'
which is the same as the force on a length 2d of one of a pair of dislocations in an infinite medium. As r grows beyond d the force falls rapidly: for r = d it is about one third of the value for an infinite medium, for r = 2d about one twentieth. Thus like dislocations in a thin film can be made to approach each other to within a distance of about d under the influence of very moderate forces. In electron micrographs a collection of like screw dislocations in a foil often seem to form a fairly regular lattice with a spacing of the order of the film thickness [64]. This is what one would expect of a set of entities with short-range hard repulsions pushed together by some external force. The elastic stresses which can exist in a deformed or non-uniformly heated foil will not usually be such as to provide this force, but there will evidently be an effective force tending to make the dislocations congregate wherever there is a local thinning of the foil, since there they will be shorter and their self-energies will be less. It is not hard to verify that the stress on a plane parallel to the axis of a screw dislocation in a plate can be reduced to zero by introducing into the plate a second dislocation of opposite sign which is the geometrical image of the first with respect to the plane. Consequently eq. (130) also gives the image force on a dislocation distant !r from the edge of a semi-infinite plate.
(131)
[I
2
[(~)2 K (nna). _ (~)2 K(nnao)J ntta., 2 2d nna 2 2d
(nna) K 2d 2
(mra2d o)
_
(nna 12 2d
o)
K2
)J-
(nna. 2d
1
and B; is the same expression with 12 in place of K 2 in the numerator. The energy E required to form the dislocated disk can be found by integrating !bPez over the area in which a plane through its axis intersects the disk. The result is simple if the outer radius is large and the inner radius either large or small compared with the thickness: 2
E
=
pb
4n
.
2d· i
(~)2 ao
,
= /lb2.2d.ln (_d_) , 4n
2.24a o
ao » d a o « d.
The expressions (131) have the complicated appearance one expects of the solution of a boundary-value problem for a finite cylinder. The fact that it could be obtained fairly easily stems from the circumstance that the dilatation is zero, so that we had to deal with harmonic rather than biharmonic functions. It would be much more difficult to find the corresponding solution for an edge dislocation. However, the case of an infinite plate has been discussed, though.in a disguised form. The problem of finding the elastic field of an edge dislocation passing through a
208
J. D. Eshelby
Ch.3
plate perpendicular to its stress-free surfaces is equivalent to the same problem for a line-force traversing the plate in the same way. To see this note that the stress function of eq. (60) for a dislocation (0, b) plus a force (0, -2ph) is harmonic, so that by the last expression of eq. (45), Pzz' Pzx and Pz y, are zero everywhere. Hence any plane z = const. is stress-free and the plane strain solution is itself the solution for a plate. This means that the image fields of the two singularities cancel, or that the image fields of a dislocation (b,O) and a line-force (0,2J1b) are identical. Green and Willmore [65] have studied the line-force case, and their paper makes clear how difficult an analytical solution of the problem is. They give some curves of the image field. Remote from the dislocation the elastic field in the plate will be given closely by the generalized plane stress solution ([2] p. 207). This is derived from the plane solution, eqs. (45), (48) by making the changes J1---+ J1, v ---+ v/{l + v), putting pzz equal to zero, and interpreting Pxx' P yy, Pxy' u and v as denoting merely averages across the thickness of the plate, though in fact their values will be nearly constant across it. Very close to the dislocation line, the generalized plane stress solution, interpreted as an average or otherwise, is incorrect: for example v will have nearly the value given by eq. (57) with the original, not the modified, Poisson's ratio. From the plane strain solution one can also derive a so-called plane stress solution ([2J p. 206) which satisfies the equilibrium equations and leaves the surfaces of the plate stress-free. However, for the edge dislocation it is useless because it contains unwanted stresses of order r" 3 near the singular line. The discussion in the last paragraph suggests that the total force exerted by one edge dislocation in a plate on another is the same as it would have exerted on a length 2d of the latter if they were both in an infinite medium. When the dislocations are very close or very far apart compared with the plate thickness the result follows from the assumed validity of the plane strain solution close to and the generalized plane stress solution remote from one of the dislocations, but strictly speaking one can say nothing precise about the case when their distance apart is of the order of the thickness of the plate. The stress field of the dislocation, and with it the interaction between two dislocations, may be changed radically if the plate can relieve its stresses by buckling. The problem has been studied by Mitchell and Head [66]. They conclude that for a circular plate the condition for buckling is R > 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
2 1 2 (1 - 2 pI cos R - pi R
tp
+ 2 pi: cos 2
_ ... )
of which only the first term survives the
dV _ - 2 - 2n r
J
f11: fa 0
0
Ilb 2 1 W=8n 2 r2
and the interaction energy with the sphere is, similarly, with the same F,
E =
~: (/12
- /1,)F
(i)
(135)
with an error of order (1l2 - Ill)2; any difference in the Poisson's ratios only makes itself felt in the next approximation. Equation (135) ought to agree with Willis et al. 's [75J series solution if in it we put III = u everywhere except in the over-all multiplier u - Ill' our III - 1l2' but it is not easy to check this. At large distances, where F(a/R) = a 2/3R 2 , eqs. (134) and (135) agree with the results of Weeks et al. [72J to the stated accuracy. 3.4. Approximate methods
For the other term we set up spherical polar coordinates p, 9,
r2 - R 2
211
For a screw dislocation eq. (133) is replaced by
where W(Il, v) is the energy density in a homogeneous material with elastic constants u, v, and the integration extends over the inclusion. For an edge dislocation with the field of eq. (57) we have 1 1 (I W(Il, v) = zbD 2n r 2
Boundary problems
§3.3
sin 9 p 2 dp as R 2 - p2 sin? 9
In most of the cases we have considered, the problem of annulling the stresses on all the surfaces of a finite solid is shirked by allowing the material to extend to infinity in one or more directions. The exception eq. (131), suggests that in any case such a complete solution would not be very informative. However, useful results can be obtained for certain finite bodies if we are prepared to tolerate the degree of approximation associated with the theory of the strength of materials and use the intuitive methods employed in it. A useful tool [81, 82J in such calculations is Colonnetti's theorem [13]. Stated in words it says that the work done in introducing a dislocation into a stressed body is equal to the work done by the loads in moving through the displacements produced by the dislocation. As a first example consider an edge dislocation passing through a rectangular rod
= 4naF(R/a) with F
= 1 - x- l(1 -
X 2)1/2
sin- l x.
Hence the interaction energy takes the form
E ibD, '2a{LF(~) + M~: cos 219]. =
(134)
where the values of Land M can be inferred from eqs. (133) and (132). If, for example, matrix and sphere have shear moduli Ill' 112 and the same Poisson's ratio v we have v M = 3(1 _ v) L.
Fig.6 (a) An edge dislocation passes through a rod with couples applied to its ends. (b) The area over which the product of fibre stress and Burgers vector must be integrated. (c) The dislocation is not straight, or forms a closed loop.
Ch.3
J. D. Eshelby
212
(or plate) as shown in fig. 6a, b. Obviously the rod will be un strained except near the dislocation, but because of the local deformation there the ends will be rotated through some angle f3. To find it bend the rod by couples M applied to the ends. The work done by the dislocation, if it is introduced from the top, is the fibre stress induced by the couples, integrated over the shaded area in fig. 6b and multiplied by the magnitude of the Burgers vector, which we suppose parallel to the axis of the rod. Elementary beam theory gives the value 3bM(c 2 - y2)j4c 3 for the work, where 2c is the depth of the beam and y is the height of the dislocation above the mid-plane. The work done by the couples is f3M, the product of the couple and the relative rotation of the ends. Equating these two quantities we have f3 = ib(c 2 y2)jc 3 (Kroupa [83J, Siems et al. [84J). Some similar results have been given by Siems et al. [82J. The same method applies if the cross-section is not rectangular and the dislocation is not straight, or if it forms a closed loop, fig. 6c. There will then be bending about each of the principal axes of the cross-section and they may be found by applying a corresponding bending couple and equating f3M to b times the integral of the fibre stress over the appropriate shaded area. If the dislocation deforms by gliding on a cylinder with the shaded area as base it does no work against the applied stresses and f3 is unaltered. Thus an arbitrary dislocation in the rod produces the same total bending as its projection on a cross-section of the rod. For example, suppose that half the dislocation in fig. 6b glides towards the reader so as to form an axial screw dislocation joined to the surface by edge segments at its ends. The total bending is still the same, but it now occurs in two equal instalments at the exit points. The two planes of bending will not usually be parallel because of the twist induced by the intervening screw segment (see below). A component of the Burgers vector not parallel to the axis of the rod does not affect f3. In the anti-plane solutions for a screw dislocation discussed in sect. 2.2. there is a net couple, M say, about the z-axis acting on each cross-section of the material. Consequently if a finite rod is cut out of the infinite cylinder to which the solution applies the rod will develop a twist unless appropriate couples are provided. For a screw dislocation at the centre of a circular cylinder we find M = pba' and so, dividing by the torsional rigidity inf-UI4, we get the value
i
o:
= bjna 2
§3.4
Boundary problems
usually circular, and the dislocation is not necessarily at the centre. We therefore need to know the twist due to a dislocation at (x, y) in a whisker whose cross-section is bounded by an arbitrary closed curve C. It can be found at once if the appropriate torsion function lJI(x, y) is known. The torsion function [19J satisfies
V2 'P = -2 'P
= 0
inside C on C
and in terms of it the torsional rigidity is the integral D = 2/1
f
'P dx dy
taken over the interior of C. If the whisker is given a twist rx l by applying end couples M = rx l D the stresses in it are
r.. =
Jirx l 8'Pj8y,
and they exert a force
Fx = bpzy = - ub«, 8'Pj8x, Fy = -bpzx = - pb«, 8lJ1j8y on an axial screw dislocation in the rod, so that 'P acts as a potential function for the force. If the dislocation moves in from the boundary to (x, y) the work done on it is
-
f
(F• x dx
+ F y dy) =
brxlJi
f8'P fu ds
per unit length of the whisker, taken along any path, or simply ba ; JilJl(x, y) since
'P is zero on the boundary. The work done by the end couples is .All«, where rx is the twist actually produced by the presence of the dislocation and M 1 is the couple associated with the externally applied twist rx l . By Colonnetti's theorem these two amounts of work are equal and we have
z(x, y) = 'P(x, y) ub]D.
(137)
Ifwe write
(136)
radians per unit length for the twist (Mann [85J). There will, of course, be end effects which can in principle be made out from eq. (127). The relation between the signs of the dislocation and the twist can be described as follows. On the surface of the undislocated cylinder scribe a generator and a set of circles with spacing b. When the dislocation is introduced the circles join up to form a helix and the generator is twisted into another helix, of much coarser pitch. The two helices are of opposite hand. A similar rule applies to the cases which follow. Twists of the order predicted by eq. (136) have actually been observed in crystal whiskers containing an axial screw dislocation, but the whisker cross-section is not
213
rx
=
Burgers vector area of cross-section
K--------
K is close to unity for a dislocation near the centre of a reasonably equiaxed crosssection. For a dislocation at the centre of an equilateral triangle, a square or a regular hexagon it has the values 10/9, 1.048, 1.015 respectively. It is also unity for a dislocation at the centre of an ellipse of any excentricity. In contrast it has the value i for a long thin rectangle with the dislocation anywhere on the centre line not too near the ends; this follows from the torsion function,
(138)
Ch.3
J. D. Eshelby
214
for a narrow rectangle with its long sides at y = ± id. If the dislocation has a hollow core we must use the torsion function for a suitable hollow rod. Equation (137) still applies, with P given its (constant) value on the inner boundary. (We may imagine that a singular dislocation moves in from the surface until it falls into a ready-prepared hole.) If the hollow core is nearly as large as the cross-section, so that the whisker becomes a thin tube, not necessarily of constant thickness, the torsional rigidity is very nearly D = flPI(A o
+
AI)'
where A o ' AI are the areas within the outer and inner boundaries of the tube and PI is the value of P at the inner boundary [19]. If we were interested in the torsion problems we should have to estimate PI' but it cancels from eq. (137) to give a
=
bj(A o
+
§3.4
Boundary problems
215
a curvature when the end couples necessary to maintain a state of plane strain are relaxed, but this is not so because in fact there are no such couples ([13] p. 274). To see directly that there is no curvature, bend the whisker by end couples and introduce the edge dislocation. The fibre stresses due to the couples exert no forces on it, it does no work and, by Colonnetti's theorem, neither do the couples. Hence the ends remain parallel and there is no curvature. By applying simple tension instead of a bending moment we can show similarly that the edge dislocation does not change the .Iength of the rod. The energy per unit length of a long rod containing an edge dislocation is therefore given by the unmodified plane strain formula. For a circular rod to which eqs. (75) or (86) or a suitable combination of them applies there is no stable position for an edge dislocation. For a mixed dislocation free to glide along a diameter of a
(139)
AI)
or, since AI must be nearly as large as A o for eq. (139) to apply, say simply a
=
w
ibjA o .
(Actually the more elaborate expression of eq. (139) is exact for a tube bounded by any pair of concentric ellipses with parallel axes and the same excentricity.) Thus a solid dislocated whisker with K of order unity loses about half its twist when hollowed out to form a tube. As long as the whisker is maintained in a state of anti-plane strain by suitable end couples its elastic energy per unit length is given by eq. (37). When the couples are relaxed the energy is decreased by ia 2 D, where a is the twist due to the dislocation and eq. (37) has to be replaced by E(x, y)
JJb2 [ In cP(x, y) - S 2n P 2(x, y) ] , = 4;
where S = Djfl is the geometrical torsional rigidity. For a circle the original energy is given by eq. (14), P is i(a 2 gives
(140) Fig. 7 Energy of a screw dislocation in a circular cylinder as a function of its distance from the axis. -
r
2
)
and eq. (140)
(141) which is plotted in fig. 7. Surprisingly the dislocation is in metastable equilibrium at the centre, being trapped in a potential well of depth flb 2 In (ie)j8n and radius [1 - ~(2)l/2] l/2 a = 0.54a. It can be dislodged by applying a couple large enough to undo half the twist, or by clamping one end and deflecting the other ([13] p. 273). The E(x, y) surface is similar to that of eq. (141) for other reasonably equiaxed cross-sections, and one can show that the dislocation is stable at the centre of any regular polygon. From eqs. (43) and (138) it can be shown that it is unstable in a long thin rectangle; as the cross-section elongates from a square instability probably starts when the ratio of the sides is about 2: 1. It has been suggested, by analogy with the foregoing, that a rod containing an edge dislocation parallel to its axis will exhibit
circular cylinder and having its Burgers vector inclined at an angle e to the cylinder axis the energy is given by cos? e times that of eq. (141) plus sin? e times eq. (75). The potential well of fig. 7 becomes shallower with increasing eand disappears when tan eexceeds (1 - V)l/2 [86]. Gomer [87] observed that when a whisker twisted by a screw dislocation was stretched the ends rotated relative to one another. The writer [88] gave a theory of the effect. The angle of rotation, De say, is given by
De = aolo(A2 - /12)(8 - /) fl 2S
+ (A2
-
fl2)/
(142)
where aD is the initial twist due to the screw dislocation, A is the ratio of the final length to the initial length 10 of the whisker, fl2 is the ratio of the final to the initial area of the cross-section, while 8 is the torsional rigidity for unit shear modulus and!
216
J. D. Eshelby
Ch.3
the polar moment of inertia about the centroid. Equation (142) was derived from the theory of small deformations superposed on large, so that although the dislocation strains are small A, and u are finite. The result does not depend on the details of the finite stress-strain relations, except in as far as they determine the relation between A, and u. If Hooke's law can be applied to the extension eq. (142) becomes
M) =
20: 0(1
+
v)(1 - I/S) bl
(143)
where bl is the increase in the length of the whisker. The writer claimed [89J, incorrectly, that eq. (143) showed that one could detect the presence of the internal stresses set up by the dislocation by measuring the response of the whisker to external forces, contrary to a general result in the linear theory of elasticity [90J. The presence of the dislocation gives a prismatic whisker a certain observable initial twisted shape. A rod, cast, forged or turned from the solid to the same twisted form, and free of internal stress, would also show the effect predicted by eq. (143). Thus nothing can be learned from the tensile test which could not be inferred directly from the external form of the whisker. (In the only case where the initial twist cannot be observed, that of a circular cross-section, S is equal to I and there is no effect.) To see this, without using the theory of the extension of rods with initial twist, we may start with the fact, evident from the analysis of ref [89J, that eq. (143) applies equally well when 0: 0 is the total initial twist produced by a set of dislocations distributed in any way over the cross-section. If, in particular, they are infinitesimal and uniformly distributed over the cross-section we have simply a plastically twisted rod with no macroscopic internal stresses but with the same response to axial loading as it had when it contained a single dislocation which produced the same initial twist 0: 0 ,
Boundary problems
217
where C is a suitable cap bounded by the dislocation line L. Let the loop be changed to a slightly different one L' in which every element dL of L has undergone a small displacement b~m which adds (or subtracts) a surface element d.S'.J = e,jmn bY;Sm tn dL to the edge of C, converting it to the cap C' bounded by L'. The change in Eim results from the change in the region of integration and from the change in P~j :
se;
=
ib j
~ ±b
j
r P~j dS + ib Jcr bpL dS Je-c P\hm. O~mt. dL + ±b, Ie op\j dS j
i
L
j
(146)
j .
The first term in eq. (146) evidently gives half the force predicted by eq. (144). The question is, does the second term provide the other half? The following is a compressed and somewhat distorted version of Gavazza and Barnett's careful argument to show that the second term in eq. (146) is indeed equal to the first. It may be checked with the help of Gauss's theorem, the equilibrium equations and Hooke's law that the integral
Is (Piju; -
P;jU,) dSj
(147)
involving two elastic fields u., Pij and u;, P;j has the same value on S = S 1 as it has on S = S2 if S 1 can be deformed into S2 without passing through singularities. Take for the dashed and undashed quantities the elastic fields of the final and initial loops L', L; then by expanding S until it becomes the outer free surface where p·.n. and p;jnj are both zero we see that the integral term (147) vanishes. If we split the two fields into image and infinite-medium terms we get IJ
I
Addendum (1976) Gavazza and Barnett [91J have looked at an interesting general point relating to image forces. The Peach-Koehler formula gives [13J (144) for the force per unit length acting on an element of a dislocation loop, with Burgers vector b, at a point where its tangent is the unit vector tn' due to its interaction with an externally-applied stress Pu- It is commonly assumed that if for a dislocation in a finite solid with stress-free surfaces we wish to find the force which its own image stress P~j exerts on it we have only to put Pij = P~j in eq. (144). This is plausible on the anthropomorphic grounds that the dislocation does not discriminate between externally-exerted and self-produced stresses in its neighbourhood, but it is not hard to raise doubts. The part of the elastic energy which depends on the image field is
s.; ~ ±b
j
Ie pL as,
(145)
Is (u;ropu -
u'('p;j) as, +
Is (u;'pu -
';'P;j) dS j
+
Is (u;Aj -
u,p;\) dSj
=
o.
(148)
The first integral is ofjust the same form as (147) and so its surface may be deformed into a large sphere of radius r; this is allowed since the infinity quantities are supposed to exist everywhere. As the remote displacement and stress of a loop are proportional to r" 2 and r" 3 the integral is proportional to r" 3 and yet independent of r, and so zero. The second and third integrals taken together are of the form of the term (147) and can be contracted to a jacket enveloping a surface C" which contains both C and C'. Ultimately the jacket becomes the two sides of C" plus two narrow tubes containing Land L'. It can be shown that the tubes contribute nothing because any volume element traversed by a dislocation is in static equilibrium. As the integrand is continuous across C " the rest of the second integral is also zero. In the final integral, as U i is continuous except across C and h; except across C', and the discontinuities
Ch.3
J. D. Eshelby
218
are both b, we have
b,
j' P:j dS
j
-
C'
Putting p;~ = P~j
ib i
+
hi ( P:} as,
Jc
=
O.
bp~j and dividing by two we get
r P~j dS Jcr bp~j as, = ib Je-c i
j
= ib i
f P~j
Cjmn
«.; dS
j ,
Boundary problems
219
explained by Yoffe [61J, angular dislocations can be added together to produce polygonal loops, dislocation tetrahedra and so forth. The authors reach their solution by introducing an image angular dislocation which at the site of the proposed free surface cancels the shear stresses and doubles the normal stress (compare the argument following eq. (110)) and then liquidate the latter by adding the field, eq. (121) with a suitably chosen harmonic cp. A problem mentioned in sect. 3.3., the interaction between a screw dislocation and a spherical inhomogeneity, has now also been treated by Gavazza and Barnett [103].
L
as required. Generalizing one of the results in sect. 2.3., Vitek [92J has given the Muskhelishvili complex stress functions for the elastic field of an edge dislocation near an elliptical hole in an isotropic medium. Of special importance is the case where one axis of the ellipse is allowed to dwindle to zero, leaving a dislocation in the neighbourhood of a crack. This configuration, already touched on in sect. 2.4, is important in connection with crack-tip plasticity (Riedel [93J gives useful references to the fracture mechanics literature). Vitek [94J gives the crack limit of his ellipse solution, and the same problem has also been treated by Hirth and Wagoner [95J who allow their dislocation to have both screw and edge components, and by Rice and Thompson [96J who give the image force on the dislocation. It has also been treated for an anisotropic medium. Many years ago Stroh [97J gave expressions for the elastic field of a crack in an anisotropic medium in the form of integrals involving the traction which the applied forces produced along the proposed site of the crack before it was introduced. It has been applied to the dislocation case by Atkinson [98J, Atkinson and Clements [99J and Solovev [100]. The complementary case of a dislocation near what in fracture mechanics would be called an external crack has been discussed by Tomate [101]. Here instead of, say, the segment Ixl < a of the x-axis being cracked it is the only part of the x-axis which is not cracked; in other words cracks extend inwards from - CfJ to -a and from 'XJ to a. Tomate only considers a screw dislocation, but he allows the half spaces y > 0 and y < 0 to have different elastic constants, since his real interest is in imperfectly bonded composites. A useful addition to the stock of three-dimensional special solutions is Comninou and Dundurs' [102J expression for the elastic field of one of Yoffe's [61J angular dislocations when it is situated in a semi-infinite solid with a stress-free surface. The vertex is an arbitrary distance below the surface, the two infinite arms of the angle point away from the surface and one of them is perpendicular to it. This is not really a restriction, for if we add together the elastic fields of a pair of such angular dislocations with a common vertex and the same Burgers vector the arms which are perpendicular to the surface will be coincident but described in opposite senses, so that their contributions to the elastic field cancel, leaving the field of a single angular dislocation with no non-geometrical limitations on the orientations of its arms with respect to the free surface or each other. Similarly if we add Yoffe's [60J solution for a straight dislocation in a half-space, discussed in sect. 3.1., we can cancel one arm of the angle and arrive at an angular dislocation with an arm which reaches the surface. By repeating this manoeuvre we can arrange that both arms reach the surface. As
References [1] M. Gebbia, Annali Mat. pura appl. [3] 7 (1902) 141. [2] A E. H. Love, A Treatise on the Mathematical Theory of Elasticity (Cambridge University Press, Cambridge, 1927). [3] T. A Cruse, Int. J. Solids Struct. 5 (1969) 1259. [4] J. D. Eshe1by, Phil. Trans. Roy. Soc. A244 (1951) 87. [5] P. Coulomb and J. Friedel, in Dislocations and Mechanical Properties of Crystals (1. C. Fisher et al., eds.) (Wiley, New York, 1957, p. 555). [6] A K. Head, Phil. Mag. 44 (1953) 92. [7] J. Dundurs, in Mathematical Theory of Dislocations (T. Mura, ed.) (AS.M.E., New York, 1969) p. 70. [8] J. D. Eshelby, Phil. Mag. 3 (1958) 440. [9] Z. Nehari, Conformal Mapping (McGraw-Hill, New York, 1952). [10] M. v. Laue, 1. Radioakt. Elektronik 15 (1918) 205. [11] H. Kober, Dictionary of Conformal Representations (Dover, New York, 1957). [12] G. Leibfried and H.-D. Dietze, Z. Phys. 126 (1949) 790. [13] F. R. N. Nabarro, Theory of Crystal Dislocations (Clarendon Press, Oxford, 1967). [14] G. B. Spence, J. Appl. Phys. 33 (1962) 729. [15] R. Siems, Phys. kondens. Materie 2 (1964) 1. [16] A. G. Greenhill, Q. JI. pure appl. Math. 15 (1878) 10. [17] L. M. Milne-Thomson, Jacobian Elliptic Function Tables (Dover, New York, 1950, p. 35). [18] A. G. Greenhill, The Applications of Elliptic Functions (Macmillan, London, 1892, p. 292). [19] I. S. Sokolnikoff, Mathematical Theory of Elasticity (McGraw-Hill, New York, 1956). [20] V. Volterra, Annls scient. Ec. norm. sup., Paris [3] 24 (1907) 401. [21] V. Volterra and E. Volterra, Sur les Distortions des Corps Elastiques (Memorial des Sciences Mathematiques, fasc. 147) (Gauthier-Villars, Paris, 1960). [22] O. M. Corbino, Nature (Lond.) 90 (1910) 540. [23] Ph. Frank and R. v. Mises, Die Differential- und Integralgleichungen der Mechanik und Physik (Riemann-Weber), Part I (Vieweg, Brunswick, 1925) p. 635. [24] W. F. Brown, Phys. Rev. 60 (1941) 139. [25] J. S. Koehler, Phys. Rev. 60 (1941) 397. [26] A. Seeger, in Encyclopedia of Physics (S. Fliigge, ed.) vol. 7 part 1 (Springer, Berlin, 1955) pp. 560-3. [27] J. Dundurs and G. P. Sendeckyj, J. Mech. Phys. Solids 13 (1965) 141. [28] J. Dundurs and T. Mura, J. Mech. Phys. Solids 12 (1964) 177. [29] R. D. List, Proc. Camb. Phil. Soc. 65 (1969) 823. [30] K. Aderogba, Proc. Camb. Phil. Soc. 73 (1972) 269. [31] A. K. Head, Proc. Phys. Soc. (Lond.) B 66 (1953) 793. [32] W. S. Burnside and A. W. Panton, The Theory of Equations (Longmans, London, 1912) p. 167. [33] L. A. Pastur, E. P. Fel'dman, A. M. Kosevich and V. M. Kosevich, Sov. Phys. Solid State 4 (1963) 1896. [34] A. K. Head, Phys. Stat. solidi 10 (1965) 481. [35] J. Gemperlova, Phys. Stat. solidi 30 (1968) 261.
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J. D. Eshelby
Boundary problems
Ch.3
M. O. Tucker, Phil. Mag. 19 (1969) 114I. J. D. Eshelby, Br. J. appl. Phys. 17 (1966) 113I. M.-S. Lee and J. Dundurs, lnt. J. Engng. Sci. 11 (1973) 87. B. A. Bilby and J. D. Eshelby, in Fracture (H. Liebowitz, ed.) (vol. 1, Academic, New York, 1969) p.99. [40] A J. Ardell and R. B. Nicholson, with an Appendix by J. D. Eshelby, Acta Metall. 14 (1966) 1295. [41] J. 1. Musher, Am. J. Phys. 34 (1966) 267. [42] R. Weeks, J. Dundurs and M. Stippes, lnt. J. Engng. Sci. 6 (1968) 365. [43] T. Kurihara, lnt. J. Engng. Sci. 11 (1973) 89I. [44] J. A Steketee, Can. J. Phys. 36 (1958) 192. [45] T. Maruyama, Bull. Earthquake Res. lnst. 42 (1964) 289. [46] R. D. Mindlin, Physics 7 (1936) 195. [47] R. D. Mindlin and D. H. Cheng, J. appl. Phys. 21 (1950) 926. [48] H. M. Westergaard, Theory of Elasticity and Plasticity (Harvard, Cambridge, Mass., 1952) p. 142. [49] A 1. Lur'e, Three-dimensional Problems in the Theory of Elasticity (lnterscience, New York, 1964) p.132. [50] L. Solomon, Elasticite lineaire (Masson, Paris, 1968) p. 541. [51] E. Melan, Z. angew. Math. Mech. 12 (1932) 343. [52] E. Melan, Z. angew. Math. Mech. 20 (1940) 368. [53] T. K. Tung and T. H. Lin, J. appl. Mech. 29 (1966) 363. [54] D. J. Bacon and P. P. Groves, in Fundamental Aspects of Dislocation Theory (J. A. Simmons et al., eds.) (Spec. Pub. 317, vol. 1, Nat. Bur. Standards, Washington, 1970) p. 35. [55] P. P. Groves and D. J. Bacon, Phil. Mag. 22 (1970) 83. [56] N. J. Salamon and J. Dundurs, J. Elasticity 1 (1971) 153. [57] J. D. Eshelby, in Progress in Solid Mechanics (1. N. Sneddon and R. Hill, eds.) vol. 2 (North-Holland, Amsterdam, 1961) p. 119. [58] J. Bastecka, Czech. J. Phys. B14 (1970) 702. [59] J. D. Eshelby and A. N. Stroh, Phil. Mag. 42 (1951) 140I. [60] E. H. Yoffe, Phil. Mag. 6 (1961) 1147. [61] E. H. Yoffe, Phil. Mag. 5 (1960) 16I. [62] T. Vreeland and J. D. Eshelby, in Dislocations and Mechanical Properties of Crystals (J. C. Fisher et aI., eds.) (Wiley, New York, 1957) p. 91. [63] J. D. Eshelby, C. W. A Newey, P. L. Pratt and A. B. Lidiard, Phil. Mag. 3 (1958) 75. [64] W. J. Tunstall, P. B. Hirsch and J. Steeds, Phil. Mag. 4 (1964) 511. [65] A. E. Green and T. J. Willmore, Proc. Roy. Soc. Al93 (1948) 229. [66] L. H. Mitchell and A K. Head, J. Mech. Phys. Solids 9 (1961) 131. [67] L. Rongved, Proc. Second Midwestern Conference on Solid Mechanics (1953) 1-13. [68] J. Dundurs and M. Hetenyi, J. appl. Mech. 32 (1965) 671. [69] C. F. Hsieh and J. Dundurs, lnt. J. Engng. Sci. 11 (1973) 933. [70] F. R. N. Nabarro, Phil. Mag. 42 (1951) 1224. [71] P. Coulomb, Acta metall. 5 (1957) 538. [72] R. W. Weeks, S. R. Pati, M. F. Ashby and P. Barrand, Acta metall. 17 (1969) 1403. [73] R. Bullough and R. C. Newman, Phil. Mag. 7 (1962) 529. [74] J. R. Willis and R. Bullough, in Proc. Brit. Nuc. Energy Soc. Conf. Reading (S. F. Pugh et al., eds.) (AE.R.E., Harwell, 1971) p. 133. [75] J. R. Willis, M. R. Hayns and R. Bullough, Proc. Roy. Soc. A329 (1972) 121. [76] W. D. Collins, Proc. Lond. Math. Soc. 3 (1959) 9. [77] W. D. Collins, J. Lond. Math. Soc. 34 (1959) 343. [78] W. D. Collins, Z. angew. Math. Phys. 11 (1960) 1. [79] V. 1. Blokh, Teoriya Uprugosti (University Press, Kharkov, 1964) p. 450. [80] J. H. Bramble, Z. angew. Math. Phys. 12 (1961) 1. [81] J. D. Eshelby, Phys. Stat. solidi 2 (1962) 1021. [82] R. Siems, P. Delavignette and S. Amelinckx, Phys. Stat. solidi 3 (1963) 872. [83] F. Kroupa, Czech. J. Phys., B9 (1959) 332, 488.
[84] [85] [86] [87] [88]
221
R. Siems, P. Delavignette and S. Amelinckx, Phys. Stat. solidi 2 (1962) 421. E. H. Mann, Proc. Roy. Soc. A199 (1949) 376. J. P. Hirth and F. C. Frank, Phil. Mag. 3 (1958) 1110. R. Gomer, J. Chern. Phys. 28 (1958) 457. J. D. Eshelby, in Growth and Perfection of Crystals (R. H. Doremus et al., eds.) (Wiley, New York, 1958) p. 130. [89] J. D. Eshelby, in Proc. Symp. Internal Stresses and Fatigue in Metals (G. M. Rassweiler et al., eds.) (Elsevier, Amsterdam, 1959) p. 4I. [90] R. V. Southwell, Theory of Elasticity (Clarendon Press, Oxford, 1936) p. 77. [91] S. D. Gavazza and D. M. Barnett, Scripta Metall. 9 (1975) 1263. [92] V. Vitek, J. Mech. Phys. Solids 24 (1976) 67. [93] H. Riedel, J. Mech. Phys. Solids 24 (1976) 277. [94] V. Vitek, J. Mech. Phys. Solids 24 (1976) 263. [95] J. P. Hirth and R. H. Wagoner, lnt. J. Solids Structures 12 (1976) 117. [96] J. R. Rice and R. M. Thompson, Phil. Mag. 29 (1974) 73. [97] A N. Stroh, Phil. Mag. 3 (1958) 625. [98] C. Atkinson, lnt. J. Fracture Mech. 2 (1966) 567. [99] C. Atkinson and D. L. Clements, Acta Metall. 21 (1973) 55. [100] V. A. Solovev, Phys. Stat. Sol. 65b (1974) 857. [101] O. Tomate, lnt. J. Fracture Mech. 4 (1968) 357. [102] M. Comninou and J. Dundurs, J. Elasticity 5 (1975) 203. [103] S. D. Gavazza and D. M. Barnett, lnt. J. Engng. Sci. 12 (1974) 1025.
CHAPTER 4
Nonlinear Elastic Problems B. K. D. GAIROLA lnstitut fur Theoretische und Angewandte Physik Unioersitdt Stuttgart, FR Germany
© North-Holland Publishing Company,
1979
Dislocations in Solids Edited by F. R. N. Nabarro
Contents 1. Introduction
1. Introduction
225
2. Nonlinear theory of elasticity
227
Description of deformation 227 2.2. Transition to curvilinear coordinates 231 2.3. Geometric meaning of the strain tensors 236 2.4. Principal axes of strain and strain invariants 238 2.5. Deformation of area and volume 241 2.6. Compatibility conditions 243 2.7. Stress 249 2.8. Equations of motion and equilibrium 252 2.9. Constitutive equations 254 2.1.
2.10. Material symmetry restrictions on the strain energy function and the elastic constants 263 3.1. Discrete dislocations in an elastic continuum 263 3.2. The stresses and strains around a dislocation 265 3.3. Dimensional changes in crystals caused by dislocations 287 3.4. Continuous distribution of dislocations 294 3.5. Relationship between the dislocation density and the incompatibility 297
3. Nonlinear elastic problems in dislocation theory
3.6. Determination of the stress and strain fields due to a given dislocation density Appendix: Tensor analysis in curvilinear coordinates 324 AI. Vectors and tensors in terms of natural base vectors 325 A2. Covariant differentiation 328 A3. Physical components 331 Addendum (1976) 334 References 340
302
260
The utility of the linear theory of elasticity in solving problems of dislocation theory is amply demonstrated in other chapters. Nevertheless, there are a number of situations in which it is inadequate. For instance, close to the dislocation core strains become very large, and it is obvious that the linear theory can no longer be valid. Another example is the effect of dislocations on the macroscopic density of crystals. The linear theory of elasticity predicts a vanishing effect, although it has been known for a long time that dislocations lead to a positive volume expansion [1-3]. Therefore, it is necessary to develop a nonlinear approach to deal with such problems. The earliest method of this type is due to Peierls [4J and has been further developed by Nabarro [5J, Seeger and Schiller [6, 7]. However, this method does not employ the nonlinear theory of elasticity in the usual sense, since the elastic properties in one plane only (the glide plane) are treated nonlinearly. This chapter is written with the aim .of providing a comprehensive account of the methods which employ the nonlinear theory of elasticity in treating dislocation problems. Apart from the volume expansion due to dislocations we have considered only the more fundamental problem of determining the stress and strain fields due to dislocations. This choice was dictated by the requirement of holding the chapter to a reasonable length. The methods described here have however been applied successfully to a number of other problems, such as the scattering of elastic waves by straight dislocations and kinks and the small-angle scattering of X-rays by dislocation lines and rings [8-10]. The nonlinear theory of elasticity is considerably more complicated than the usual linear theory. Already the problem of characterizing the strain is more difficult. Several kinds of finite-strain definitions have been used, which fall into two classes: definitions in terms of the undeformed configuration and definitions in terms of the deformed configuration. The linear theory, on the other hand, is in many ways too degenerate; in the limit of small deformations too many basic distinctions, effectsand difficulties disappear. Since our objective was to make this chapter as far as possible self-contained, we have provided in sect. 2 all the necessary background in nonlinear theory of elasticity that is needed in sect. 3, where the aforementioned methods of handling the dislocation problems are described. For readers desiring a more elaborate account of this subject there exist several excellent books [11-17]. It is assumed that the reader is familiar with the concepts of vectors and tensors and their representation in the Cartesian system. A brief account of selected parts of tensor formalism in curvilinear coordinates is given in the appendix. The material in sect. 3 is, for the sake of organizational convenience, arranged according to two different types of situations in the continuum theory involving either
226
B. K. D. Gairola
Ch.4
isolated dislocations in an elastic continuum or continuous distributions of dislocations. Of course, the former may also be considered as a limiting case of the latter situation. However, there is an important difference. In the former case there exists a well-defined displacement field outside the singularity so that the usual displacement function methods can be used. Such methods have been applied by Seeger and coworkers [18-20J to problems in dislocation theory. In the latter situation the compatibility equations are not satisfied so that the formalism based on displacement-gradient is no longer applicable. This difficulty is bypassed by the elegant formalism of Kondo, Bilby, Kroner and their associates [21-25]. In this method one replaces the deformation gradients ofelasticity theory by the elastic distortions which, in general, are not associated with any displacement field and one assumes that the unstressed state of the dislocated body is realizable in an appropriate non-Euclidean space. Within the framework of this modern approach Kroner and coworkers have solved a number of problems. Their method of solution employs stress functions for which a fourth-order partial differential equation is obtained by using the fundamental geometric equation which relates dislocation density with incompatibility of strains. In spite of the elegance of the differential geometric formalism it has a drawback which has not yet been overcome. This is because the stress function is described in the coordinates of the deformed body. However, the symmetry of the crystal exists only in the undeformed state and not, in general, in the deformed state because deformation depends not only on strain but also on rotation of an element with respect to the crystallographic axes. Attempts have been made by several authors to formulate the differential geometric theory in terms of un deformed state coordinates [26-28]. However, the physical interpretation of the quantities involved is rather difficult, and the application of these approaches to physical problems remains to be explored. Hence we have excluded them from our considerations. The complications mentioned above are avoided by Willis [29J by using a more elementary method which makes no use of non-Euclidean geometry. In this approach one solves the equations for the distortions themselves. A systematic description of this method has also been given by Teodosiu and Seeger [30J. In all these methods mentioned above we do not solve the nonlinear partial differential equation directly. We rather employ perturbation techniques. A calculation consistent up to second order is, generally, sufficient since little is known about the elastic constants of higher order. To some extent the approach of Seeger and Wesolowski [19, 20J is an exception because it partially avoids the perturbation treatment. These authors use the concept of controllable deformations which they call universal solutions. Such solutions make it possible to treat the nonlinear elasticity problems without prior specification of the strain energy function or the stressstrain relations and also provide a theoretical basis for the experimental determination of the stress-strain relations. Therefore, this approach has a considerable advantage over the perturbation method mentioned above. In an isotropic medium the strain field of a screw dislocation has rotational symmetry which means a great deal of mathematical simplification. For this reason the methods for determining the stress and strain fields due to dislocations are in
Nonlinear elastic problems
§1
227
nearly every case illustrated using the example of a straight screw dislocation in an isotropic elastic medium. Moreover, this practice is followed by almost all the authors. Only Kroner and coworkers [23-25J have also considered edge dislocations.
2. Nonlinear theory of elasticity 2.1. Description of deformation The deformation of an elastic body can be conveniently described in the following way. We define the position of any material point of the body in the initial state by its radius vector R which has the components Zl' Z2 and Z3 in a rectangular Cartesian coordinate system, i.e. R = ikZk where t. are the base vectors of the coordinate system. The initial state of the body may be any arbitrary state. However, in the following we shall take the initial state to be the natural state, i.e. the stress-free undeformed state to which the body would return when it is unloaded. Let us consider some particular point at R at time t = O. During the deformation it is displaced to a new position r with the components Zl' Z2' Z3 at time t. We shall assume that the change in the configuration of the body is continuous, i.e. neighbourhoods are changed into neighbourhoods. Any introduction of new boundary surfaces such as is caused by tearing or fracture of a test specimen must be regarded as an extraordinary circumstance requiring special treatment. The deformation is then described by the relation r
=
r(R, t)
(1)
or
which has the unique inverse R
= Rir, t)
or
Z, =
Zk(Z l ' Z2' Z3'
t)
(2)
for every point in the body. The functions r(R, t) and Rtr, t) are assumed to be singlevalued and differentiable with respect to their arguments up to as high an order as we wish. When a body deforms the distances between its points change. Let us consider two neighbouring points at Rand R + dR. The square of the distance between them is given by dS 2 = dR· dR = 6k1 az, az; After the deformation they occupy the positions rand r distance is now ds 2 = dr- dr = 6kZ dz, dz..
(3)
+ dr. The square of the (4)
The relationship between the deformed infinitesimal vector dr and the un deformed vector dR is given by dr = dR· VH , = rVH·dR
(5)
dr = dR· AT = A·dR,
(6)
or
B. K. D. Gairola
228
Ch.4
where VRr means the directional derivatives of I' defined as
_ . 8(ijz) _ 8zj . . VR r - 'k 8Z 'k'j
--az-k -
(7)
1=7
_ -
j
8(ijz j) . _ 8zj . . -8-- lk - -8 ljlk'
z,
Thus the arrow on V indicates that the differential operator acts on the preceding quantity. The tensor (9)
A = rVR = Ajkiik'
= det (A jk) = det (8z/8Zk) =f. O.
dS 2 = dr·h·dr = h.Jm dz J dz m
(21)
ds 2 = dR· H ·dR = H jm dZj dZm,
(22)
i
and
where or
where
A jk = 8z/8Zk (10) is usually called the tensor of deformation gradient and provides a primitive measure of deformation. Similarly we have dR
=
dr Vr R
=
RVr .dr
(11)
or
dR = dr- ( A - 1)T = A - 1 . dr,
(12)
where
(20)
We now see that both dS 2 and ds 2 may be written in another way, namely
(8)
z,
229
The assumptions we have made imply that the Jacobian of transformation does not vanish, i.e.
k
and AT means transpose of A. Obviously rVR is the transpose of the tensor VRI', i.e.
rVR
Nonlinear elastic problems
§2.l
h jm
- 1A - l = A kj km
(23)
and (24)
or
The tensors hand H are, respectively, called Cauchy's deformation tensor and Green's deformation tensor. Their reciprocals are given by h- 1 = A· AT or (25) ti:' jm = A jk A mk> - 1 = A-1A-l H- 1 = A-1·(A-1)T (26) H jm or jk mk :
It follows that the change in the square of the distance between the two neighbouring points may also be written in two ways, viz. (13)
and
ds 2
-
dS 2 = 2 dR· E·dR = 2E.jm d.Z.J dZm
(27)
ds 2
-
dS 2 = 2 dr- e- dr = 2ejm dZj dz m,
(28)
or (14)
with
where
Ai:/ = 8Zk/8z j.
(15)
In terms of components eqs. (6) and (12) can be expressed in the form
2e = I - h
dz , = A jk dZk,
(16)
az, =
(17)
Ai:/ dz j.
Obviously
A 1·A=A·A-1=1
or
Aji/Akl=AjkAi:/=bjl'
(18)
where I, the identity tensor, also called the unit tensor, is defined by the requirement that l·u=u·l=u
for all vectors u.
2E = H - 1
or or
2Ejm = H jm - bjm'
2ejm = bjm - h jm.
(29)
(30)
The tensors E and e are called Lagrangian and Eulerian strain tensors, respectively. They are more often used as deformation measures. It may be noticed that all of the tensors H, h, E and e are symmetric, i.e. H = H T etc. Besides these many other measures of deformation have been proposed by various authors. For instance Hencky's logarithmic measures [31J are defined by E = tlog H = tlog(1
+
+ ~E3 + e 2 + ~e3 + ....
2E) = £ - £2
e = - t log h = - t log (1 - 2e) = e
(19)
Some recent papers on various strain measures are those by Karni and Reiner Seth [33J and MacVean [34].
(31) (32)
B. K. D. Gairola
230
Ch.4
Nonlinear elastic problems
§2.1
231
where -
1
E. = 2:('V R U
+
t=;UV R)
(39)
and
e=
t(V'ru
+ uVr )
(40)
are called the linear Lagrangian and Eulerian strain tensors. In the classical theory of linear elasticity one assumes that the displacements themselves are infinitesimal so that
V'RU = A - I « I
or
V'r u = I -
A-I
« I
(41)
and it follows that (42)
Thus it is immaterial whether the derivatives of the displacement are calculated at the position of a point before or after deformation and so the distinction between the Lagrangian and Eulerian strain tensors disappears. Fig. 1. Displacement of two infinitesimally separated material points during deformation.
2.2. Transition to curvilinear coordinates If we introduce the displacement vector u = r - R (see fig. 1) with components Uk = Zk - Zk' then A
=
A-I
+ uVR
I
= 6 jk + 8u/8Zk, A jk I = 6 jk - 8u/8zk,
or
= 1- uVr
A jk
or
(33) (34)
and the strain tensors E. and e take the form (35.1)
If the body has curvilinear boundaries it is, in general, more convenient to use curvilinear coordinate systems for the different states of the body. This makes the imposition of boundary conditions easier. A brief introduction to the requisite parts of tensor analysis in curvilinear coordinates is given in the appendix at the end of this chapter. To include all coordinate systems commonly used one should admit transformations of the form (43)
or
=
E. jm
1. [8U j 2
8Z
8um
m
8uk 8uk
+ 8Z.J + 8Z.J 8Z m
J'
(35.2)
These transformations are arbitrary except for the restriction that their Jacobians be positive. Using eqs. (1) and (20), we can eliminate Zk from eq. (43) and obtain
x" = Xk(XK , t).
(44)
In the same way we get x K = XK(X k, t).
(45)
and (36.1)
or
ejm = t[8U j + 8um _ 8uk 8Uk] 8zm 8zj 8zj 8zm
It also follows that •
(36.2)
If we assume the displacement gradients to be so small that the terms involving their products can be neglected in comparison to the displacement gradients themselves we can write ds 2
-
dS 2 = 2 dR· E·dR =
2 dr- e·dr,
(37) (38)
j
= det (8x kj8X k ) =I- O.
(46)
It is not necessary for X K and x k to be measured with reference to the same coordinate system. In fact they may be two different frames in relative motion; the two frames could be specified by giving two rectangular Cartesian reference systems moving as rigid bodies relative to each other. This general scheme, in which the choice of the two coordinate systems is independent was introduced by Murnaghan [35J. Figure 2 shows a body in its natural state in a rectangular Cartesian system Z K with unit base
B. K. D. Gairola
232
Ch.4
Nonlinear elastic problems
§2.2
233
We can also define the reciprocal base vectors G K, Ok and metric coefficients G KL, gkl as shown in the appendix. We now have k _ :v - oXorK G K -_ oX ox or K_ K ox k G -
A - r
R -
k K A KOk G ,
(54)
where oxkjoX K.
=
A\
(55)
We thus find that A has the components A\ along the base vectors at two different points. This is an example of two point tensor functions [36]. Similarly we can see that
=
A-I
RV,.
=
A-I\GKOk,
(56)
where Z2
Fig. 2.
A- I Kk
Coordinate systems in the natural state and the deformed state of the body.
oXKjoXk.
=
(57)
Obviously vectors IK and in the deformed state at time t in the rectangular Cartesian system Zk with unit base vectors ik . In the natural state, curvilinear coordinates X K with covariant base vectors GK are shown, while in the deformed state another set of curvilinear coordinates x k may be chosen with covariant base vectors Ok' Since the position of a material point in the natural state is now given by curvilinear coordinates X K or by a vector R, we have
A\A- I KI
dR = GK dX
K
,
GK
=
A· GK
=
orjox\
=
<5 JK ·
(58)
=
A\Ok'
Ok. A
=
A\G K
(59)
and A-I'Ok
(48)
and (49)
The material point in the deformed state is represented by a position vector r or curvilinear coordinates x k referred to another coordinate system whose base vectors are given by Ok
A-IJkAkK
=
G K. A-I
A-IKkGK,
=
(60)
A-I\Ok.
Therefore, the components of Cauchy's and Green's deformation tensors are given by
where GK are the base vectors of the coordinate system given by oRjoX K.
s-;
Furthermore
(47)
Hence we can write
=
(50)
so that
hkl = Ok·(A- I ? A-I'O I = GKLA-IKkA-ILI
(61)
H KL = GK· AT. A· GL = gkIA\A IL.
(62)
The displacement vector u which extends from a material point in the undeformed body to the same material point in the deformed body is now given by u
=
r - R
+ p,
(63)
where p is an arbitrary constant vector. The displacement vector can be expressed in terms of components along any of the base vectors GK' Ok' G K and Ok. For instance if this vector has the component UK along G K and Uk along Ok we can write
(51) 2£KL
These two coordinate systems with their base vectors are also shown in fig. 2. The metric coefficients of the two curvilinear coordinate systems are GKL
and
=
GK· G L
= H KL - GKL =
(G)\7 K U L·+ (G)\7 L UK
2ekl = 9 kl - h kl = (g)\7 kUI (52)
+
J + (G)\7KUJ)(G)n V LV '
(g)\7 I Uk - (g)\7kUj
r-v U vI
j
,
(64) (65)
where the covariant derivative (g)\7k ( ) are defined with the help of connections (g)r~ in the same way as (G)\7k( ) with the help of (G)r1jfL in the appendix. Until now we have considered the coordinates x" of a point of the deformed body referred to a fixed coordinate system. We can also proceed in another way and
B. K. D. Gairola
234
Ch.4
identify a point of the deformed body by means of the coordinates X K which define the initial position of the point before the deformation. This is equivalent to the assumption that X K are the curvilinear coordinates in the deformed body. These coordinate lines are derived from the original coordinate lines by supposing them to be carried along by the body in its motion and deformed with it. (One can visualize the coordinate lines as coloured elastic threads frozen in a transparent medium.) Thus the coordinates of a point in the undeformed as well as in the deformed state keep numerically the same values. Alternatively we can proceed by identifying a point of the undeformed body by means of the curvilinear coordinates x k which are derived from the coordinates in the deformed body by supposing them to be carried along by the body in the reverse motion as the deformed body relaxes to the natural state, i.e. the stress-free undeformed state. Both of these procedures are demonstrated in fig. 3. However, for the sake of clarity, the coordinate systems at the start of each procedure are rectangular Cartesian.
§2.2
Nonlinear elastic problems
235
Thus we can write dr and dR in the form dr = H K dX K ,
(68)
dR = h k dx".
(69)
We also get H KL =
hkl =
HK·HL
hk·h l
=
=
I gkIA\A L,
(70) (71)
GKLA-1\A-1LI'
The vectors H K and hk reciprocal to H K and hk are given by
=
H K
=
hk
!(H)fKLMH
!(h)fklmh
l
X H
L
X h
(72)
M,
(73)
m,
and (h)fklm are defined in the same way as Substitution of eqs. (66), (67) and also
where(H)fKLM
(G)fKLM
in the appendix. (A.19)
ti,
and (74)
in eqs. (72) and (73) yields H K
J2
h
1,
k
A- 1K kU k
= =
AkKG
K
=
G
K.
A- 1 ,
= s': A.
(75) (76)
We see clearly that
Fig. 3.
H K
(b)
(a)
Embedded coordinate system which coincides with a fixed Cartesian system in (a) the natural state and (b) the deformed state of the body.
We shall see later on that the description of deformation from the above point of view has certain advantages, particularly because in this description the body can be considered as a material space that is in certain cases more general than the ambient space. The base vectors H K which form the curvilinear coordinate system X K in the deformed body are given by H K =
or oXK =
k ox or oX K oxk =
k A KUk'
(66)
Similarly the base vectors of the curvilinear coordinate system x" in the undeformed body are hk =
oR
-0k x
=
ax K en
-0 k 0 x
X
K =
A-
1\G K·
(67)
#-
GKLH L,
(77)
= H- 1K L H v
(78)
but rather H K
where H- 1K L
h-
1kl
=
=
HK.H L
hk·h
l
=
G
= KL
gkIA-1\A-1LI' I
A\A L
(79) (80)
are reciprocal to H KL and hkl . We may now write eqs. (54) and (56) in the form A =
HKG
A -1 =
K
hkU
k
= Ukh\ = GKH K .
(81) (82)
From the present viewpoint then we can consider H K L and H- 1K L as the metric coefficients of the deformed material space, i.e. the point space which we identify with the deformed body when we use the moving coordinate system X K and in the same way hkl and h- 1kl as the metric coefficients of the material space in the natural
B. K. D. Gairola
236
Ch.4
state in the moving coordinate system x". This can be seen by expressing the unit tensor in the form I = HKLHKH L = hk1hkh l = H-1KL[IKHL = h-1klhkhl'
§2.3
(83) (
H = HKLGKG L,
(84)
h = hklgki
(85)
h- 1
=
1kl
(KK)
)1/2 = (1
+ A(K»(I +
A(L» cos () - cos
e
(no sum).
(94)
(LL)
e
For K = L we have 8 = 2EKK / G (KK) = (1
A(K)
(86)
H-1KLGKGL'
h-
2EKL )1/2 (G
G
+
= 0; therefore we get 1
A(K)2 -
(no sum)
(95)
or
and their reciprocals by
=
237
where 8 is the angle between the base vectorsHK andHL and e is the angle between GK and G L • In view ofeqs. (91) and (92) we can write eq. (93) in the form
Note, however, that Green's and Cauchy's deformation tensors are given by
H- 1
Nonlinear elastic problems
(87)
gkgl'
We can, therefore, interpret the strain components from two different points of view. In one case it is the difference between the components of Cauchy's (or Green's) deformation tensor and the unit tensor in the same coordinate system. In the other case it is the difference between the metric coefficient before and after the deformation in a coordinate system which moves and deforms with the medium. Some authors use the so-called convected coordinates. These are obtained from the general scheme of coordinates we have used here by assuming X K = x". In these coordinates components of G KL and h kl coincide and so one can write G k1 = h kl and obtain ds 2 - dS 2 = (gkl - G dx" dx'. (88)
= (1 + 2EKK / G (KK» 1/2
1
-
(no sum).
(96)
In Cartesian coordinates G(KK) = 1 and if we assume E KL to be small we getthe usual result of the geometrically linear theory A(K)
~
EKK
(no sum).
(97)
Let us now consider the change in angle between two vector line elements dR 1 and e 12 between them is given by
dR 2 · The angle
dR 1 ·dR 2 = IdRll·ldR21 cos
e 12
(98)
or (99)
k1)
where N l and N 2 are the unit vectors. The angle 8 12 between the deformed elements dr 1 and dr 2 is similarly given by
2.3. Geometric meaning of the strain tensors In order to see the geometric meaning of the strain tensors let us first consider the change in length of a line element dR. If we denote by A the change in length per unit length of the vector dR so that
dr 1 dr 2 cos 812 = Idr 1' 1l'ldr2
(100)
In view of eq. (90) we get Idrl - IdRI ds - dS A = IdRI = dS
(89) cos 812 = (1
we have Idrl = (1
+
(90)
A)IdRI·
We call A the elongation of dR and we see from eq. (89) that the elongations in the directions of the base vectors GK are given by
A(K)
= (1 + A(K»IGKI· However,IHKI = (HK ·H K ) 1/2 =
(91)
IHKI
(H(KK»1/2
= (1 +
(H(KK»1/2
A(K»(G(KK»1/2
and
IGKI
=
(G(KK»1/2,
(no sum on K).
=
H KL -
G KL
IHKI·IHLI
=
HK·HL -
so that (92)
e,
1 A(l»(1
dr 2
+
A(2» IdR 21 T
- - - - - - - N · A ·A·N
(1
+
+
A(2»
1
2
1 (G KL
+ 2EKL)N1K N 2 L,
(101)
or using eq, (99) 1 cos 812 = (l + A(l»(1 + A(2) (cos
e 12
+ 2E KLN1K N 2 L).
(102)
In the special case when dR 1 and dR 2 are perpendicular to each other we get
G K· G L
cos 8 - IGKI·IGLI cos
+
A(l» IdR l I (1
(1 + A(1»(1 + A(2»
We now write the definition of the strain tensor E KL in the form 2EKL =
dr 1
(93)
. 2E KLN1 K N 2 L 12 sma = (1 + A(l»(l + A(2»,
(103)
B. K. D. Gairola
238
Ch.4
IIh = det f. = .g.[2 tr f.3
where 0: 12
=
e12
6>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
~ o.
(158)
2u
a a ax J ax M - ax M ax J
2
au
axJ axM
a2 u
axM axJ
a_
= O.
a u M J ax ax
(159)
As V K are the components of a vector which has been spread into a field by parallel transport, it is easily seen that the actual change in V K on proceeding from one point to the next is equal to the change on parallel transport as expressed by eq. (A.36). Hence we can put
avL = _
(G)r L
ax J
JK
VK
in eq. (159) and thus obtain (G)RJMKLVK
= 0,
(161)
where ::l(G)r L (G)
L
_
R JMK -
U
MK
ex!
(162)
Since eq. (161) holds for arbitrary V K we must have (163) We show in appendix A.2 that the changes in the components ofa vector or tensor on parallel transport are entirely due to the changes in the base vectors. Therefore eq. (163) can be looked upon as the integrability condition of the equation dGK =
(G)ryKGL dX J,
(164)
K
K
G
0
= C'VJ'VMU - 'VM'VJU ) K = .
(165)
= R JMLK U LGK = 0,
(166)
which again yields the condition of eq. (163). This equation also demonstrates that R JMKL are the components of a tensor which is usually called the Riemann-Christoffel tensor or simply the curvature tensor. It is shown in appendix (A.2) that (G)r ~K
reduce to the form of eq. (A.54) if eq. (157) is satisfied. Hence we can write eq. (162) as (G)RjMK
(160)
= O.
Using eq. (A.40) we get 2
_ _ «G)rL VK) axM .JK
=
2 K 2 K (a u a u ). M axJ axM - ax ax J 'K
However, two vectors which are equal in one system of coordinates would be equal in all systems of coordinates. Hence we should have
Applying Stokes's theorem we obtain the equivalent equation
~ «G)r L V K) axJ MK
245
L
= a{ktK}(G)jaxJ - aUK}(G)jaX M + UN }(G){~}(G) - {ktN hG){~K}(G)
(167)
The starred symbol R denotes that part of the curvature tensor which is formed only by Christoffel symbols. We can now characterize the compatibility of deformation which carries the body from the natural to the deformed state in the following way. The deformation is compatible if a compact body remains compact in the deformed state too. Geometrically this is equivalent to the statement that the deformation is compatible if the material space in the deformed state is also Euclidean. That means in the first place that the embedded coordinate system which has deformed with the body should remain holonomic so that the position vector r of the material point in the deformed state is a continuous single-valued function of these coordinates. Hence we must have (168) In view of Stokes's theorem it follows that (169)
B. K. D. Gairola
246
Ch.4
which implies that there exists a function r such that AT = VRr.
(170)
=
x
V R X
(GKHK)
=
G
J
X
G
K
;~~ +
G
J
x
~~:HK'
SinceH
=
oHJ oX K
(H)r L JK
(H)rL)H KJ L
_
= 0
(H)r L
JK ".
KJ -
(172)
.
(173)
+
oHJN _ oXK
OHJ K) oX N
L M JK
= 0
1 (OHK L
=2
[JK, L] (H)
oXJ
+
oHJ L oXK- -
OHJ K) oX L
=
N {JK}(G)HN L
+
2EJ K L,
E JK L
=
!(G)VJEK L
=
HLN[JK '
N]
(H)
+
(G)
VKEJ L
-
(182)
(G)V LEJK)'
= {L }
JK (H)'
=
(G)RjMKL
+
2[(G)V JEM K L
(174)
(G)V MEJKL -
-
(175)
2HNP(EJLPEMKN
EMLpEJKN)]
R:
1 JK
L
= 0,
(H)RjMKL
=
2[(G)VJEM K L -
(G)V MEJKL -
2H
NP(EJLp
E MKN -
(176)
J
=
[JK, LJH)
+
(183)
EMLpEJKN)]
= O.
where (H) R MJK L and (H) R MJK * Lare defined in terms of (H)r Land {L} in the same JK . JK (H) way as (G)R M J K L and (G)Ri;JK L in eqs. (162) and (167). A more convenient form of eq. (176) is obtained by lowering the index L and using the relation OHKL/OX
= O.
However, we started with the assumption that the material space in the natural state is Euclidean. Hence we must pu~ (G)RjMKL equal to zero. We thus have
which in view of eq. (174) reduces to (H)
(181)
Substituting eq. (181) in eq. (178) and rearranging the terms we get
The second condition which must be fulfilled in order that the material space in the deformed state be Euclidean is (H)R
(179)
KN
where
0.
.1HLN(OHKN 2 oXJ
{NJL } (G)H
Hence we can write
(H)RjMKL
=
L -
(180)
It follows that (H)r~K reduce to a form similar to eq. (A.54), i.e. (H)r L JK
{N} JK (G)HN
OHK L oX J -
(171)
Lare independent we get the condition
(H)r L
_
KL -
The second term vanishes due to the symmetry condition of eq. (A.53). Therefore we obtain oHK _ oXJ
247
procedure given by Kroner and Seeger [23]. We first note that in view of eqs. (64), (A.44) and (A.54),
(G)VJH
That means the mapping R ----+ r = r(R) is continuous and single-valued. If we substitute eq. (81) in eq. (169) we get VR
Nonlinear elastic problems
(184)
We can easily deduce the following symmetry properties (H)R* JMKL -
(H)R* MJKL -
-
-
(H)R* JMLK -
(H)R* KLJM'
Therefore, as shown in subsect. 2.5, we can introduce a tensor dual to the curvature tensor given by i(H )fQJM(H )fRKL(H)RJMKL
[JL, K](H)'
(177)
= G (G)fQJM(G)fRKL(G)VJEMKL
We get
-
2H
NP
EJLPEMKN)'
(185)
H
(H)R* JMKL -
H
LN
(H)R* N JMK
This tensor is usually known as the Einstein tensor. It follows that the compatibility condition can also be expressed in the equivalent form
_ o[MK, LJ(H) -
oX J
n*QR
(178) This equation can be expressed in terms of the strain E K L by using eq. (64). A great many authors have given such equations in various forms which are, often, rather complicated. A recent paper on this topic is that of Bondar [43]. References to older literature are to be found in Truesdell and Toupin [15]. We prefer to use here the
= -
(IncRE)QR -
2(G)fQJM(G)fRKL H N P EJLPEMKN
= 0,
(186)
where (IncRE)QR
=
-(G)fQJM(G)fRKL(G)VJEMKL -
(G) fQJM (G) fRKL(G) V J (G)V K EM L
= (17R
x E
X VR)QR.
(187)
Ch.4
B. K. D. Gairola
248
The tensor IncR E. is called the incompatibility of E.. It was introduced by Kroner in the linear continuum theory of dislocations [44J. It is appropriate to consider 17* as a nonlinear generalization of it. Hence we shall refer to 17* as the Lagrangian incompatibility tensor. It is easily verified that, when the coordinate system is Cartesian and when squares and products of the derivatives of EJK's can be neglected, eq. (186) reduces to 8
2
+
ELM
8
2
EJ K
8
EJL
8
2
EK M = 0
8z J 8z L
8z K 8z M
8z L 8z M
8z J 8z K
2
(188) .
These are the well-known compatibility conditions of infinitesimal strains which were first given by Kirchhoff [45J, St. Venant [46J and Boussinesq [47]. The explicit forms in curvilinear coordinates were given by Odqvist [48J and Vlasov [49]. Let us now consider the compatibility of deformation which carries the body from the deformed to the natural state. We again assume that the deformed body is compact, i.e. the material space in the deformed state is Euclidean. Clearly, we can proceed in the same way as before and characterize this deformation as compatible if the natural state is Euclidean also. That means, firstly
249
Nonlinear elastic problems
§2.6
From eq. (65) and (g)Vjg kl = 0 it follows that .
[jk, IJ(hl
1 (8h kl
= 2"
+
8x j
8h j l 8xk -
8h j k) 8x l
=
n Lk} (g)h nl -
(196)
2ejkl'
where (197) Hence we finally obtain (h)R* jmkl -
(glR*
jmkl
-
2[(g)n e vj
mkl
-
(gln e v m jkl
+
2hnp(ejlpemkn -
emlpejkn)J
= O.
(198)
Since we started with the assumption that the material space in the deformed state is Euclidean we can put (g)Rjmkl equal to zero. The resulting equation can be reduced to the equivalent form n*qr
= (Inc, e)qr
-
~qr
= 0,
(199)
where (189)
(Inc, e)qr = -
-(g)fqjm(glfrkIV.Ve
which implies
VI'
X
]
(A- 1)T = O.
(h)r l kj -
= (V I'
(190)
0,
(192) Therefore
=
ek}(h)
=
(193)
him [jk, mJ(hl'
Secondly, the curvature tensor of the material space in the natural state must vanish, i.e.
»«.jmk I =
8(hlr I
8(hlr I
8x]
8~
~
jk
+
(h)r~ (h)r n jn
mk
_ (hlr I (hlr~ mn]k
= 0
.
(194)
In view of eq. (193) this equation may be put in the equivalent form (hlR* jmkl -
e x
ml
VI' )qr
(200)
(201)
(191) The tensor
(h)rjk
X
k
and
Substitution of eq. (82) in eq. (190) leads to (h)r l jk
(g)fqjm(glfrkl V.e ] mkl
h (hlR* n In jmk
_ 8[mk, IJ(hl 8x j
(195)
17*
may be called the Eulerian incompatibility tensor.
2.7. Stress From the continuum viewpoint the forces which act on the body may be classified as the body or volume forces which act on the mass points of the body, the surface forces which act on the bounding surface of the body, and the internal forces acting across surfaces between two parts of the body. We shall assume that a force density per unit mass exists so that the total body force is obtained by integrating over the volume of the body. Similarly we shall assume that a surface force per unit area exists. According tothe stress principle of Cauchy, internal forces acting across surfaces in the interior of a body are assumed to be of the same kind as the distributed surface loads. Imagine a closed surface s within the body. The material exterior to this surface interacts with that in the interior. We can say that the forces exerted by the material on one side of the surface s are equivalent to a distribution of stress vector t having the dimension of force per unit area. Consider a small surface element of area ~s on our imagined surface s. We can distinguish the two sides of & according to the direction of a unit vector n normal to ~s. The part of the material lying on the positive side of the normal exerts a force /1P on the other part which is situated on the negative side
B. K. D. Gairola
250
Ch.4
of the normal. We shall assume that the vector I1P/l1s tends to a definite limit dP/ds as /1s tends to zero. The limiting vector is the stress vector t given by t
= dP/ds
dP =
or
t
ds.
(202)
The forces with which the two parts act on one another cannot give anything but zero in the total resultant force, since they cancel by Newton's third law. Consequently the stress vectors acting at the same point r but on opposite sides of a surface are equal in magnitude and opposite in direction, i.e. t(r, n)
= - t(r, - n).
(203)
The total surface force on the portion enclosed by s can therefore be regarded as the sum of the forces exerted by the portion surrounding it. Since these forces act on the surface of that portion, the resultant force can be represented as the sum of forces acting on all the surface elements, i.e. as an integral over the surface. Let us consider a small tetrahedron with one corner at r, and oriented in such a way that its three faces are the coordinate surfaces and the fourth face is I1s. In the limit I1s ---+ 0 we use the well-known result: the vector sum of the areas of the tetrahedron is zero. Hence we can write n ds
= n(1) dS(l) +
11(2)
dS(2) +
11(3)
dS(3) ,
(204) ~
where 1I(i) are the unit vectors normal to the coordinate surfaces dS(i)' Thus we have resolved ds = n ds into its components dS(i) along the three unit vectors. Substituting
§2.7
Nonlinear elastic problems
where (J is called the Cauchy stress tensor. Since it is a function of deformed state coordinates, the strain measure suitable to use with it is the Eulerian strain. However, sometimes we need to relate stresses to Lagrangian strains. Hence, it would be convenient to define the state of stress by a stress tensor at r measured per unit area of the undeformed body. For this purpose we introduce the stress vector T(r, N) referred to the un deformed area such that dP(r)
= t(r, n) ds = T(r, N) dS.
T
= N·l
t- =
or
NKL Kk
(206)
in eq. (204) we arrive at the relation dS(k)
= (g(kk»)1/2 nk ds = (g(kk))1/2 dsk.
(207)
dP = ds·(J = dS·l
(214.1)
dp k = dSj(Jjk
(214.2)
or
= dSKL Kk.
Thus we see that the tensor I = GK9kLKk is a two point tensor function which associates the force vector dP at r with vector area dS at R. From eqs. (150) and (214) it follows that dS· [J A- 1 . (J
-
I] = O.
t ds = t(k) dS(k) .
(208)
Body forces acting on the tetrahedron do not contribute to this equation, because they are of higher order of smallness than the surface forces. We see from the last two equations that t =
(i kk))1/2t(k)nk =
(209)
(1k nk.
or
(216)
9kt k
~
I is usually called the Piola-Kirchhoff stress tensor [50, 51] of the first kind. Another stress tensor which is called the Piola-Kirchhoff tensor of the second kind is formulated in the following way. Instead of the force dP(r) we consider a force dP'(R) acting on the surface dS and introduce the stress vector T'(R, N) such that
~
= nj9k(JJ = n19 1 .9j9k(JJ .
dP'(R)
= T'(R, N) dS = A -1. T(r, N) dS = A- l. ttr, n) ds.
(218)
T' = N·T.
(219)
In view of eq. (218) it follows that
= A- l·(N·l) dS = A- l·(n·(J)ds.
(220)
Therefore using eq. (150) we obtain T
or
(217)
Here again we can put
(210)
Thus we find that t=n·(1
= A- l·dP(r).
Then
N'T dS
Resolving t and (1k along the base vectors 9k we get
(215)
Therefore we must have
dP'(R)
Let the stress vector on dS(k) be denoted by t(k). The equation of motion of the infinitesimal tetrahedron is, in the limit,
(213)
and therefore eq. (212) can be written in the form
and
= 9K/lgK I = gK/(g(kk»)1/2
(212)
Proceeding in the same way as before we can put
(205)
n(k)
251
= A- l·l = l·(A- l )T = JA- l·(J·(A- l )T, lIT
(211)
(J=-A·l=-A·T·A J J
'
(221) (222)
Ch.4
B. K. D. Gairola
252
§2.8
L
or in component form
rK L =
I:,KIA L
I
=
= ~ AkKL K I = ~ A\AIL"C K L. J J
(Jkl
t
= nij
-JK _ (J -
The tensor
JK
(225)
,
A-iJA-iK jk j k(J.
(j
=
iiJKHJHK
+
253
L
(224)
We note that we can define the stress tensor referred to the deformed area in another way by using natural state coordinates. For this purpose we resolve t and n along the base vectors H K . Using the same arguments as above we obtain K
r x (1- a)pdv
(223)
J(JkIA-i\A-iLI,
Nonlinear elastic problems
r x (n·a)ds
=
L
{r
X
[(f- alp
+
V.·a]
+ O'A}dv ~
0,
(232)
where (233) Since these equations are valid for arbitrary v and since the integrands are continuous, it follows that the latter vanish identically. Hence from eq. (231) we obtain
(I - a)p + Vy ' a = 0
(226)
or
(fl -
d)p
+
(9lVk(jkl
= O.
(234)
These are known as Cauchy's equations of motion.
is called the convected stress tensor.
In the special case of static equilibrium of the medium, the acceleration a is zero and these equations reduce to the partial differential equations of equilibrium 2.8. Equations of motion and equilibrium
fp
The equations of motion can be derived from the conservation laws of linear and angular momentums. Let us consider some portion of the body which is of volume v and bounded by the surface s. Let/be the body force per unit mass and t be the surface force per unit area, or traction. From our discussion in the previous section it is obvious that the resultant force and resultant moment acting on the portion are given respectively by (227) and
L
r x fp dv +
L
r x t ds.
ap dv =
i
fp dv
+
1 i t ds =
fp dv
+
1
I'» + (9lVk(Jkl =
or
0
O.
(235)
If the surface forces are prescribed, these equations are supplemented by the boundary conditions. In view of eq. (234) we obtain from eq. (232) the condition or
(Jkl -
(Jlk
= 0,
(236)
that is, the Cauchy stress tensor a is symmetric. The equations of equilibrium in the natural state coordinates can easily be obtained by using the Piela-Kirchhoff stress tensors. Using eq. (222) we just replace (J by the equivalent expressions in terms of Piola-Kirchhoff stress tensors. For example, eq. (235) in terms of the Piola-Kirchhoff stress tensor of the first kind takes the form fp
(228)
+
V.·GA.I)
(237)
= 0.
In view of eqs. (81) and (149) we can put
The conservation of linear momentum and angular momentum implies
i
+ Vy·a =
(238)
n- 0' ds,
(229)
Furthermore, by differentiating the determinant G we get { r x ap dv = { r x
Jv
Jv
Ip dv + { r Js
x t ds
= { r x Ip dv + { r x (n- a) ds,
Jv
Js
(230) where a is the acceleration vector. Making use of Gauss's theorem we can write these equations in the forms
L
[(I - alp
+ V.' a] dv
= 0,
(231)
dG = GG K L dGK L
(239)
or d log G l /2 = tG K L dGK L
.
(240)
Taking into consideration eq. (A.44) we obtain l 2 ~ ax J log G /
=
(Glr L JL'
(241)
Ch.4
B. K. D. Gairola
254
In a similar manner we can get
~ log H l / 2 aJ(J
=
(Hlr L
(242)
JL'
From these results and eqs. (A.53) and (173) it follows that
Vr .(~ J
A)
=
~ (G)r L 2J JL
(Glr L LJ
(H)r L
JL
+
(H)r L
)G J = O.
LJ
(243)
Therefore, eq. (237) reduces to
[Po
+
A· Vr·I = [Po
+ VR·I =
0
(244)
or
flpo
+ VK:E KI =
0,
k
a
a
Equation (236) on the other hand assumes a more complicated form (A·I)A = 0 or A\:E KI - AIK:E Kk = 0,
)
(246
Nonlinear elastic problems
255
sistent with the accepted physical laws. Therefore we shall follow here the usual thermodynamical approach which is more familiar to physicists. F or our purpose it is sufficient to consider only the special situations when the change of state from the undeformed state to the deformed state is reversible and either adiabatic or isothermal. In either of these cases one can show from thermodynamical considerations that there exists a strain energy density per unit mass which is an analytic function of strain, and no explicit display of temperature in this function is necessary. Let us consider some deformed body which is maintained in the state of equilibrium by the body forces [and surface tractions t. Let u be the displacement vector of a typical material point P. We now suppose that u' is another kinematically possible displacement vector which differs from u by a small amount bu. Thus the variation
bu = u' - u
(245)
where we have used eq. (155), an equation similar to eq. (A.56) for I and the relation
A K ax k = aJ(K'
§2.9
(250)
or the virtual displacement of P is an arbitrary vector in the neighbourhood of P. We consider the variations only in the deformed state and hence we shall assume that the variations of vectors and tensors associated with the natural state are zero. For example, bGK, bGKL etc. are zero since the points in the undeformed state are not varied. The strain tensor EKL was defined by 2EKL = H KL - GKL and hence
bEKL = tbHKL = t(HK·bHL
(247)
+ H L ·bHK).
(251)
However
it
which shows that I is in general not symmetric. In rectangular cartesian coordinates the equations of motion, eqs. (234) and (244) are simply d)p + aakljaxk = 0 (248)
bHK = b fK
a,
.r -
and
au )
+ aJ(K =
au au' au b aJ(K = axK - axK
abu
k
/
= aJ(K (u - u) = axK = A Kg Vk bu:
(252)
Substituting this expression and eq. (66) in eq. (251) we get
(249)
(253) or
2.9. Constitutive equations The equilibrium equations are not sufficient to determine the stress distribution, even when the boundary tractions and body forces are given, since there are only three equations for six independent unknown stress components. To make the problem determinate, it is necessary to introduce constitutive equations or stress-strain relations defining the nature of the materials under consideration. The constitutive equations which characterize a material may be developed in several ways. The modern approach which has been developed in the last two decades starts with constitutive equations of a very general nature, and specializes the equations as little as possible and as late as possible. The theory is purely mechanical and makes no use of energetic or thermodynamic concepts. A detailed account of this approach is given by Truesdell and Noll [17]. Without the imposition of further restrictions, the constitutive equations developed in this way may, as Rivlin [52] has remarked, be incon-
t('\\bu/
+
Vlbu k) = A-1KkA-1LlbEKL'
(254)
The virtual work done by [and t can be written as
bA
=
i i
f-oup dv
+
i
(255)
t-bu ds.
By substituting eq. (211) and using Gauss' theorem we obtain
bA
=
[(fp
+ Vr · .J·bu +
o
>-
Vr bu] dv
(256.1)
or (256.2)
B. K. D. Gairola
256
Ch.4
In view of the equilibrium condition eq. (235) and the symmetry of (Jkl this equation can be put in the form bA =
!
1««.
bUI
+
v, bu,) d».
(257)
i
by! = by - T oS - S bT,
ijI'L bEn dv,
bU = bA
(258)
and the substitution in this expression for by yields (268)
+
(260)
bQ = IpTbSdV,
by dm =
1
where we have used the fact that the mass elements are conserved in a virtual displacement, i.e. b(p dv) = O.
(262)
p by dv
~
1 ss 1 pT
dv
+
ifKL
s».. dv.
(263)
We suppose that the integrands in eq. (263) are continuous functions and since the region of integration is arbitrary we conclude that by = T oS
+
1 - ii KL bEKL. p
Let us now consider the following special cases. If we assume that the body is thermally insulated so that it cannot exchange energy with the exterior (adiabatic transformation) we can put bS = O. This situation is realized when the deformation takes place so rapidly that the heat flow does not have time to equalize temperature (e.g. in vibratory motions). Thus eq. (264) reduces to the form (270)
In this case we can regard y as a function of the nine independent parameters E KL . On the other hand if we suppose that the process of deformation occurs so slowly that thermodynamic equilibrium is established in the body at every instant we can put bT = O. Therefore, we find that by! = -1 (J-KL bEKL.
(271)
P
Substitution in eq. (259) from eqs. (258), (260) and (261) gives
1
(264)
Thus we see that in either of these two cases we can define a density y or n which is independent of temperature. Since these two quantities play exactly the same role for the following calculations we will not distinguish them any more, and we will speak of an energy density y which is a function of E KL . The relation ofeq. (270) then permits us to assert that ii KL
= P oy/OEKL·
(272)
That means
This expression suggests that we regard y as a function of the independent variables Sand E. If the material is inhomogeneous the function y will also depend on the position variable. Thus we are led to consider y in the form y = y(R, S, f).
(265)
We can define a different potential function known as free energy by using the Legendre transformation
= y - TS.
=
P
(261)
pby dv,
q>
(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
b KN b L M,
(299)
+
b M Q b NP)
bMN(b K P
sLQ +
b KQ
bLP)
bPQ(b K M
bLN
+
b KN
bLM),
bKM(b L P b NQ
+
+
bLN(b KP b M Q
+
bKN(b L P b M Q
(300)
b L Q b NP)
+ +
b KQ b M P)
(301)
b L Q b M P)'
Thus the elements a(1) and a(2) constitute a tensor basis for the space of the second order elastic constants and the elements &(1), &(2) and &(3) for the space of the third order elastic constants of an isotropic material. It follows that the only independent constants of second and third order are
cg) = Cg)3
A,
= vl ,
Ci~
Ci~4
=
KL = 11 Voigt notation = 1
(302)
v2,
22 33 23 234
13
12
5
6.
All other constants are either zero or simply related to the above constants. Hence, for isotropic materials eq. (286) takes the form
-
(b)
(0)
= u,
where we have used Voigt notation according to the following scheme
Wee) = (tA
Q
+ /l)li - 2/lIIE + i(v l + 6v2 + 8v3 )Ii 2(v 2 + 2v 3 )hlIE + 4v 31lIE + ....
(303)
Let us consider next the elements of the integrity basis for the crystal classes 43m, 432, m3m in the cubic system. We now introduce these elements of table 1 into the construction of W. Differentiating the terms involving these elements and proceeding as previously we obtain three independent second order constants C 11, C 12 , C 4 4 and six independent third order constants C 11 l ' C 11 2' C 12 3' C 14 4' C 1 5 5 and C 4 56. This procedure of computing the elastic constants is quite straightforward, but the algebraic manipulations become somewhat tedious for higher order constants and materials of'lower symmetries. The number of independent elastic constants of second
Fig. 4
Burgers circuit in (a) a real crystal and (b) a perfect reference crystal.
coincident. The vector PQ required to close this circuit is defined as the true Burgers vector B. Alternatively, we may define a local Burgers vector by measuring the closure failure in the dislocated crystal of a circuit which closed in the perfect crystal [22, 67J. This procedure is shown in figs. 5a and 5b. In order to make the two definitions consistent, the closure vector Q'P' is drawn from the end point Q' to the starting point P'. These definitions can also be adopted for dislocations in an elastic continuum (Volterra dislocations). The displacement field u in a body containing a Volterra dislocation is a multivalued function. Ifwe introduce an appropriate barrier (i.e. a surface j
(a) Fig. 5
(b)
Burgers circuit in (a) a perfect reference crystal and (b) a real crystal.
Ch.4
B. K. D. Gairola
264
bounded by an open or closed curve, the dislocation line) we can consider u as singlevalued but having a discontinuity, the Burgers vector, across the barrier. The strain and stress fields are continuous, twice differentiable and finite everywhere except at the dislocation line where they are singular. This singularity may be removed by excavating a channel around the dislocation line, and thus making the body multiplyconnected. The natural state of this body can be realized by making a cut over the barrier and then scraping away material if there were interpenetration. Let us now consider a closed circuit c in the dislocated crystal (deformed state) enclosing the dislocation line in the right-handed sense. In fig. 6b this circuit with the starting point P and the end point Q coinciding on the barrier encloses an edge dislocation and hence corresponds to the circuit in fig. 4a. In the natural state the points P and Q of the same circuit are no longer coincident (fig. 6a). We join these two points and call the new closed curve c', i.e. c' = c + QP. From the integrability condition of eq. (156) it follows that
l Ye,
dR =
or
J:
dR =
1f dR +
i
i
~
dR
Nonlinear elastic problems
§3.l
Similarly, if C is the closed circuit in the un deformed material which becomes an open circuit with non-coincident starting point P' and end point Q' in the deformed material (figs. 6a and 6b), then the integrability condition of eq. (168) can be written as
l
Je
=
dr =
Jcdr + JQ'iF' dr = 0,
where the closed circuit C' = C handed sense. It follows that h
=
iF' dr =
JQ'
-ic
dr = -
(308)
+ Q' P' encloses the dislocation line in the right-
l
dR· AT.
(309)
-
fc
(310)
Jc
Using eqs. (33) and (156) we get b
~
-
fc
dR·(I
+ VBu) ~
du = -(u)e·
P
(304)
dR = 0,
Q
-
i A~l·dr.
(305)
When c and C enclose several dislocation lines, eqs. (307) and (310) define the resultant Burgers vectors of all these dislocations. The local Burgers vector b in general differs from the true Burgers vector B because b is affected by elastic strains. When the Burgers vectors are very small we can easily derive a relationship between them by taking c' and C as the closed circuit in the undeformed material. In this case the local Burgers vector is given by
Since Jjl dR defines the true Burgers vector B we can write
B
265
i (A~l)·dr ~ i dr·(A~l)T ~ i
b =
(306)
-
dr =
A-dR.
(311)
F or very small QP we can write
In view of eq. (34) and the integrability condition of eq. (168) we obtain
B = p'dr.(I - V.u)
J: J:
h;:::; A·B.
(307)
du = - (u),
where (s), denotes the jump in the value u after going once around the circuit.
(312)
Thus the local Burgers vector is just the deformed true Burgers vector. In the linear theory the circuits are made very large so that they lie in the region of infinitesimal strains. In this situation it is not necessary to distinguish between the true Burgers vector and the local Burgers vector. 3.2. The stresses and strains around a dislocation
(a) Fig. 6
(b)
Continuum definition of (a) a true Burgers vector and (b) a local Burgers vector.
In sub sect. 3.1 we have shown that in the case of a singular dislocation there exists a single-valued displacement field u(R) in the body which has a discontinuity jump B (the Burgers vector of the dislocation) around any circuit enclosing the dislocation line. It follows that we can use the displacement function methods of elasticity theory for the determination of stress and strain fields near a dislocation. In the following we shall describe two such methods. The first one is a method of successive approximations. This was essentially the technique which Seeger and Mann [18J applied for the first time to the problem of the strain field around a straight screw dislocation. We give here a more systematic account of it. The second method is due to Seeger and Wesolowski [19, 20]. To some extent it avoids successive approximations and leads to solutions that are valid in the region closer to the dislocation cote
B. K. D. Gairola
266
Ch.4
than the region of validity of those obtained by other methods. The same could be said about a recent work of Knesl and Semela [68J because they obtain third order solutions whereas by other methods only second order solutions have been obtained so far. However, they use a rather special kind of constitutive law given by Kauderer [69J which ignores the second order terms. This constitutive assumption is so restricted that it cannot give an account of even such fundamental non-linear effects as those of Kelvin and Poynting type [70, 71]. Only for certain one-dimensional problems such as flexure and torsion of rods [72J is there no difference between a general formulation and this approach. The example of torsion in a whisker due to a screw dislocation considered by Knesl and Semela falls in this category and their result is, naturally, of some interest. However, we shall not discuss this work further. Interested readers should consult the original paper. 3.2.1. Siqnorini's method 3.2.1.1. Successive approximations. In a displacement problem, we may write the governing equations in the following way fpo
+ VR·I = 0,
or
+ V,.·a = 0,
fp
(244, 235)
§3.2
Nonlinear elastic problems
In the following we shall consider both these viewpoints. Hence, in general, the boundary conditions are for example
°
N· I = on the outer surface S N· I = T on the core surface S, .
1
J
_ aW(E) _ T -
~
-
C(2)" E
T
'
1
+ 2(C(3) .. E) .. E,
E = ![VRu + uV R + (VRU)'(UVR)J, (u)c = -B.
(276,274) (289) (35) (307)
In order toobtain a unique solution these equations must be supplemented by appropriate boundary conditions. In the continuum model the body containing a dislocation is a multiply-connected body having an outer surface and an inner (core) surface. Therefore an obvious condition is that the tractions on these free surfaces should be zero. The core, nevertheless, does exert certain tractions on the boundary of the elastic materials surrounding it. These forces, on the other hand, decrease much faster with the distance from the dislocation line than the main contribution to the stress field from the line singularity. For this reason one either ignores the core altogether and takes the stress field of the dislocation as any convenient field with the correct singularity at the dislocation line and satisfying the boundary condition mentioned above, or one assumes the traction at the core surface to be zero. Physically, however, it is more appropriate to assume that the forces exerted by the core are nonzero. These forces can, in principle, be calculated using atomic or pseudo-atomic theories (see e.g. Granzer [73J and Suzuki [74J). Further work in this direction is at present being done jointly by F. Granzer and his group and C. Teodosiu*.
°
r = R
+
I
8
n u(n)(R).
(314)
n=1
°
The summation begins at n = 1 in order that r = R when 8 = in which case there is no deformation. Substitution of eq. (314) in eq. (35) leads to a power series for E E = ±{c:[VRu(1) + U(1)VRJ + 8 2[V R U(2 ) + U(2)VR + (V RU(1))'(U(1)VR)J + 8 3[V R U(3) + U(3)VR + (V R U(1))'(U(2)VR) + (V R U(2))'(U(1)VR)J + ... } = !{2c:E(1) + 8 2[2£(2) + (VR U(1))' (U(1)VR)J + 83[2£(3) + (V R U(1))'(U(2)VR) + (V R U(2))'(U(1)VR ) ] + ... }, (315) ~he.re £ is the strain tensor of the geometrically linear theory as defined by eq. (39). Similarly for A and l/Jwe obtain the expansions A
=
I
+
1
J = I -
+
uV R = I 8
tr (U(l)VR)
8
U(1)VR
+
t8
2
+ c: 2 U(2)VR + "',
[ (tr (U(1)VR))2
+
tr (U(1)VR)2 - tr (U(2)VR)J
(316)
+ (317)
Inserting the expansions (315)-(317) in eqs. (289), (276) and (274), a formal series expression is obtained forr, I and a: 00
T
=
I
00
n
8 T(n) ,
n=1
The nth terms
L(n)
I
=
I
00
8
n
I(n) ,
a
n= 1
=
I
n
8 a(n)'
n=1
etc. can be put in the form
T(n) = C(2)" E(n)
* See Addendum (1976).
(313)
In view of the nonlinear nature of the above equations it is, generally, rather difficult to obtain exact solutions in a closed form. Hence it is natural to consider the use of approximate methods. In this section we shall discuss briefly the method of successive app~oxim~t~on which has been applied to special problems. This method, first given by Signorini [75J, has been extensively considered by Italian authors. The idea behind this method is that problems of nonlinear elasticity can be solved by successive approximations taking as the first approximation the solution to a corresponding problem oflinear elasticity. Essentially equivalent methods were suggested by several other authors, among them Misicu [76J and Green and Spratt [77]. Rivlin and Topakoglu [78J have given a simple physical interpretation of the method. We assume that the position r of the particle which initially has position R can be expressed as asymptotic power series in a parameter 8 as 8 ~ 00
a=~A·T·A
267
+ T(~),
I(n) = C(2)" £(n)
+ I(~),
(318)
Ch.4
B. K. D. Gairola
268
where T~) is a function of VR T~)
=
T~)(V R
U(l)'
VR
U(l)'
U(2')"
VR
U(2)"
VR
.. ,
.. ,
VR
u(n-l)
only, i.e. (320)
U(n-l»)'
This can be seen by writing the first two terms explicitly:
T0)
= 0,
T(2)
= 2 C(2) .. D(l)
ItL
=
Itz)
= Ttz) +
0'(1)
= 0,
0'(2)
= It~) + (C(2) .. f(l))' VR U(l)
1
+ 2(1 C(3) ..
E(l») .. E(l)'
0, U(1)VR'
(321)
C(2) .. f(l)'
-
(tr
U(l)VR)
C(2) .. f(l)'
where (322)
= (VR U(l») '(U(l)VR)'
D(l)
These expressions show that in the first order approximation (classical infinitesimal theory) the three stress tensors T, I and a are indistinguishable. We now make the assumption that the volume forces and surface tractions do not possess an axis of equilibrium and can be expressed as a power series in the parameter e. A similar assumption is made for B. Thus we have
L
f =
T =
en f(n)'
L
en
t.;
B
= L
h.l)PO
-
N.l(n)
(U(l»)c
o on s {
T(n)
on
(324)
s,
=
-B(n)'
(325)
= -
B(n)'
where f(~ =
f(n)Po
T(~)
N.l(~l\lR
=
+
VR' l(~/VR U(l)'
r R U(2)"
u(1)' V R U(2)"'"
.. ,
r R U(n-l»),
V R u(n-l»)'
c>
(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
[(_J.i_ _ n)G G A + J.i R R
(1)2
Pc R5
(372)
+
C'
4R 4
(1)
+
(1 A
2)B(;) 8n 2R 2
+ A InR B 2 ) _ J.iA'p~1)2 + (A + v R
2
{
4R 4
(1)
J.i2
= 4J.i2(l _ y)2 (A + 2v1 + 2v2) (A + J.i)2 + (A + 2v2)n
This solution must be added to that of eq. (335). Therefore the first order solution is given by
- B(1) (G G z 4nR
R
+ A (C y2p~1)2
U(1) = Perc _J.i_ R R 2J.i (r; - r~) A + J.i
A'
C~ + ABti) _ A InR B 2 _ A'p~1)2)
(365)
and
+
(371)
Equations (372), (354) and (339) lead to the two equations
(364)
Q2 = P; r~ r; /2J.i(r; - r~)
_ B(1)¢ G 2n Z
(370)
where A is given by eq. (348) and
(A
Q1 = pcr~/2(A
275
The solution of eq. (370) is of the form
The boundary conditions, therefore, yield
2(A + J.i) Q1 - 2J.iQ2/r~
Nonlinear elastic problems
§3.2
~ fl -n) [Jc ~ fl (Jc + 2v
2)
+
(3fl
+
4v
3)
(Jc
2
~ fl -n) J}
(373)
for R = rc and R = re which determine the constants C~ and C~ . It is interesting to note that, in second order approximation, the dislocation produces displacements both in the R and the Z directions, an effect entirely absent from the linear approximation. We can also see that the correct solution to be taken depends critically upon the boundary conditions assumed to hold at the core boundary.
3.2.2. Seeger and Wesolowski's Method 3.2.2.1. Controllable deformations. The starting point of Seeger and Wesolowski's +J.i
/I
r-
+ n) GGJ'
(368)
where y = r~/r; and n = t;/R 2 . For the boundary-value problem in the second order approximation everything remains the same excepting that the right-hand side of the inhomogeneous differential
method [19, 20J is the concept of controllable deformations. A deformation is called controllable, for materials of a given type, if it can be supported by surface tractions alone in every material of that type. The modern development in this field started with Rivlin's work [83J, who observed that for an incompressible elastic material a number of exact and explicit solutions to boundary-value problems can be obtained. Apart from the rather trivial homogeneous deformation, the solutions found by Rivlin can be classified into four families. A fifth one has been foundas a result of work by Ericksen [84J, Klingbeil and Shield [85J, Singh and Pipkin [86J and Fosdick [87J. Of special interest to us is the family of controllable deformations which describe inflation or eversion, bending, torsion and shear of a sector of a hollow circular cylinder. Let R, ¢, Z and r, ({J, Z be cylindrical coordinates of a typical n....... L".............
276
B. K. D. Gairola
Ch.4
in the natural state and the deformed state, respectively. Then the deformation is defined by
= (d 1 R Z + dzY/z
(374)
= d 3 ¢ + d4Z,
(375)
z = d s ¢ + d 6Z,
(376)
r cp
d 1 (d 3d6
-
d4ds )
= 1,
(377)
Nonlinear elastic problems
§3.2
additional deformation specified by a small displacement u' which brings the material point to its final position r', We shall assume that u' = fV* where c is a small parameter whose square and higher powers are neglected. Similarly we shall assume that the body force and density in the final state are given byf + ef" and P + ep" respectively. We now have dr' = A' ·dR = H~ dX K .
(378)
we have only a kind of shear of azimuthal planes. As a result the planes Z = const. are deformed into the helicoidal surfaces z - d s cp = const. We can easily see that for d s = - B/2n this corresponds to the screw dislocation. A wedge dislocation is also included in this family. This can be seen by putting d1
= 8/2nd6 ,
d3
= 2n/8,
dz
= d 4 = d s = 0,
(379)
where 0 < 8 < 2n. These and other deformations belonging to the five families produce stress fields which have such a high degree of symmetry that they can always be equilibrated by a hydrostatic pressure, irrespective of the elastic properties of the material. Thus the assumption ofincompressibility brings about a great simplification in many problems, and it was the observation of this fact which stimulated the modern development offinite elasticity theory. Incompressibility is, moreover, a good approximation for many materials. Nevertheless, no real material is truly incompressible. It is, however, possible to extend the results for the incompressible materials to slightly compressible materials by means of perturbation theory for the superposition of small deformation on finite deformation since the additional strains may be expected to be small. The essential features of this theory are described in section
(380)
That means
«;
where d 1 , d z, d 4 , d s and d 6 are arbitrary constants and d 1 and d z have values such that d 1 R z + dz > 0 when R is in some interval R 1 .~ R ~ R z . This is the most extensive family of controllable deformations. If we put
277
(381)
where H~ are the basis vectors of the embedded coordinate system in the final state given by or' H'K = oXK = H K
ov*
+f
oXK'
(382)
Accordingly H~L to the first order in e is
H IeL = HIe:
-H~ ~ H
KL
+ £ (HK ' : ; : + H L , : ; : }
(383)
If we express v* in terms of components along the base vectors H K we obtain H~L
+
=
H KL
=
(H)V K
fHiL'
(384)
where H;L
Vi +
(H)V
L Vi.
(385)
Thus the total strain E~L in the final state is given by E~L
=
}(H~L - GKL )
=
}(HKL - GKL )
= EKL +
3.2.2.2.
+ !fHiL
fEiL'
(386)
3.2.2.2. Small deformation superposed on finite deformation. This theory was developed by Green, Rivlin and Shield [88], and is also described in refs. [13-15]. It provides a natural extension of classical linear theory as initially the body need no
where
longer be in a natural state and as such it is a valuable tool in treating initial stress problems and small amplitude waves in stressed bodies. Such problems may be treated in two stages. The first stage involves the solution of the elastic problem for the finite deformation alone. In the second stage we deal with quantities involving infinitesimal strains only. In our case, of course, we have an incompressible body in the first stage for which the solution for finite deformation is known. In the second stage we have small deformations, corresponding to the slight compressibility of the body, superposed on the finite deformation. Suppose that a material point originally at R is displaced to r by the finite deformation specified by a displacement u. Upon this deformation is superposed a small
Furthermore, we can define the covariant differential (H')V J( ) with respect to the final state coordinates in terms of the connection coefficients UK}(H') which are given by
(387)
{~KtH') ~ ±(H-
1 )' L N
[JK, N]{H')'
(388)
where (389)
B. K. D. Gairola
278
Ch.4
Substituting eq. (384) in eq. (388) and taking into account only terms to the first order in f we obtain
Nonlinear elastic problems
§3.2
where fK and f* K are the components along the base vectors H K. Substituting eq. (390) in eq. (399) and retaining terms only up to first order in c we obtain
(390)
+ pfL + e «HlVK,,'KL +
(HlVK"KL
where
{~M} * "ML + {~M} * "KM
+P*fL+Pf*L)=O. (391)
According to eq. (274) the stress tensor for the strained body in the final state can be written as (392) Resolving obtain
'KL (J
279
along the base vectors
(J'
H~
and substituting eq. (381) in eq. (392) we
1 aW(E')
= J' aE'
.
KL
To the first order in
1 _ _ 1/2 J' - IIlE,
_ -
E
Since the deformation R -+ r(R) produces an equilibrium stress field, VK(J'KL + P fL = 0, and it follows from eq. (400) that
(H)VK,,*KL
(lIIE
ak
+ E IIIE*) _ 1/2
'"
1(
'" J
E
1 - 2J IIIp
)
(394)
and
{~M} * "ML + {~M} * "KM + p*fL + pfH ~ O.
(401)
For isotropic materials one can also use the deformed state coordinates because it follows from the relations of eqs. (125) to (129) that the strain energy density W may be considered to. be a function of the invariants of anyone of the strain measures E, e, H, h, H -1 and h - 1. However, one has to modify the constitutive equation accordingly. For instance let us consider Was a function of I h- 1, Ilh- 1 and IIIh- 1. Then we can write eq. (281) as
(393)
we can write
(400)
r
1 oW a(h- 1 I - J a(h -1 )mn aAIK -
n
Ak
K'
(402)
Using the relation of eq. (80) we obtain 1)mn A k a(h= GMN(b m bK An + bn bK Am )A k K aAI I M N 1 N M K K
aW(E') aW(E) aE' ~ ~E KL U KL
(403)
2
a Wee)
+ E aE KL aE MN Ei/N'
(395)
k 2 aw -1 km 2 aw -1 k a 1 = J o(h - 1ym(h ) =Ja(h-1)lm(h )m'
Inserting eqs. (394) and (395) in eq. (393) we get (396) where
(J'
Therefore
Since
W = W(Ih - 1 , IIh KL
1 aW(E) =---
(397)
and
~(
2
0 Wee) E*
J aE KL aE MN MN
1,
IIlh -
(405)
1) ,
we have
J aE KL
(J'*KL =
(404)
_
~ oW(E) III 2J aE KL
E
*).
aw aw aIh - 1 a(h- 1Ym = aIh - 1 a(h- 1Ym
aw
aIIh - 1
aw
aIIIh -
1
+ aIIh - 0(h- 1Ym + oIIIh - a(h- 1)lm' 1
1
(406)
By straightforward calculations we find (398)
Thus we see that all quantities Q characteristic of intermediate state change to Q + 8 Q* in the final state. The equilibrium equation in the final state is, therefore,
(H')VK(J'KL
+ f(H')VK(J'*KL + (p + fp*)(fL + f*L) = 0,
(399)
(407)
Ch.4
B. K. D. Gairola
280
§3.2
Nonlinear elastic problems
281
B kl = ic» - gmnGknGln,
Hence
(417)
1= gklGkl, II = t(12 - gklgmllGkmGIIl) = gk1GkIIII,
aw + alll h
(IIh- 1 bml - I h_ 1 ( h - 1)m l
1
+
(h- 1)m
n(h-
1)n
l)·
(408)
I
~ [ aw (h- 1)k + aw J alhI allh-
(I _ (h- 1)k _ (h- 1)k (h- 1)m) h1 I m I
1
1
+ a~~. (IIh~.(h-l)',
-
Ih~.(h-l)'m(h-l)m, + (h-
1)'m(h- 1)m
n(h-
1)n,)
J
Consequently, eq. (409) reduces to ell = a 1 b\ + a 2(h- 1)\ + a3(h-1)km(h-1)ml
(412)
or
+ a 2(h- 1)kl + a 3(h- 1)km(h- 1)ml,
(413)
where
a2 =
2
1/2
1
(414.2)
1
(414.3)
1
ro" + lj;1 G" + lj;2Bkl,
(415)
= 2(111)1/2
aw,
tun
we have (420)
and hence (421) where g:1 =
(g)V k
vi +
(g)V
I
v:.
(422)
Accordingly we have
{J~k}
= (g')
tg' In [j k, n]
= ~(ill + e g*IIl)(Uk,
(g')
nJg)
+ Uk, f,
n](g*»,
(423)
we get
where
' '* {jk} = 1Cg'"[jk, nJ".) +
«»:», nJg)),
2 lj;1 = (111)1/2
aw ai'
2 (111)1/2
aw alI'
+ c 1*, II' = g'k1Gk11lI' = II + f 1I*, III' = g'/G = III + f 111*, I' = g~1 Gkl = I
(426.1) (426.2) (426.3)
where 11* = gkl(g*k11II
(416)
(425)
It also follows that
1* = g:IG k1,
where p
fV*
(424) '
The form of stress-strain relation that has been used by Seeger and Wesolowski employs convected coordinates. In these coordinates, as we mentioned at the end of subsect. 2.2, the components G KL and (h - 1 )kl coincide. Thus writing G kl for (h - 1 )kl and 1, II, III for I h- 1 , IIh- 1 , IIIh- 1 we may put eq. (413) in the form first given by Green and Zerna [13] (Jkl =
(419)
(414.1)
IIIh- 1
1
= 2 ow.on.
l/J2
and therefore, to the first order in
aw -a-,
(aw aw ) - - + - - Ih alhallhaw )1/2 all . h-
(1IIh - 1 )
2 a 3 = (IIIh- 1
1/2
= 2 swiet.
(409)
(411)
l •
a 1 = 2 (1II h- 1 )
(418.3)
If the points of the body are subjected to an additional deformation
and from eq. (127.3) it follows that
(Jkl = a 1gkl
III = det (gkIGkl) = giG.
l/J1
According to the Cayley-Hamilton theorem (see subsect. 2.4) we can put (h-1)km(h-1)mn(h-1)nl = Ih_l(h-1)km(h-1)ml - IIh_ 1 ( h - 1)kl + IIIh - 1 bkl , (410) J2 = IIIh -
(418.2)
For incompressible material we have III = 1 and W = W(1, II), so thatp is arbitrary and lj;1 and lj;2 are then given by
Substitution of eq. (408) into eq. (404) yields el =
(418.1)
111* = IIIgklg:l .
(427.1)
+ gk1III*),
(427.2) (427.3)
Ch.4
B. K. D. Gairola
282
These results agree with the rule that quantities Q characteristic of the intermediate state change to Q + t Q* in the final state. Obviously this rule also applies to the quantities p, ljJ i > 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
2 a2 w
III
B2
n r
B kl =
0
(431) 0
The equilibrium equation now takes the form
a" +
pfl
+
E
k
+
{~m} *
(g)V
<Jkm
(J* kl k
+
{kkm}* a'"
+ p*I' + PI*')
=
(436)
(438)
= 1,
o
2+ 4 2 20
a
2 2 w
B2
+-4n 2 r 2
(437)
B2 , 1=11=3+4 2 2
2 a2 w
2 a2 w
B - 2nr 2
,
B
n r
2 a2 w
o
It follows that
where
(g)V
B2
r 2 + -2 4n
0
o
0
0
B
2/r 2
- 2nr 2
B2 2 + -2r2 2 4n 2nr
B
(439)
In view of these relations, eq. (415) may be written in the form (J 11
o.
=
p
(B
2
+ ljJ 1 + 2 + 4n2r 2) ljJ 2 ,
(440.1)
(432) (440.2)
In the absence of body forces we have simply (440.3)
(g)V
(Jkl k
+
E
(g)V
(J* kl k
+
{kkm}* a'"
+
{lkm}* (Jkm)
= 0
.
(433) (440.4)
3.2.2.3. Example. We consider again the example of a straight screw dislocation along the axis of an isotropic cylinder of infinite length, which is at first assumed to be incompressible. The controllable deformation corresponding to this case, as we have seen in subsect. 3.2.2.1., is given by
r = R,
qJ
=
Z=
B
- 2n
+ Z.
(434)
(440.5) This stress field can always be equilibrated by an arbitrary hydrostatic pressure in the absence of body forces so long as the material is incompressible. For a slightly compressible material this is no longer true. In this case we must superpose a small deformation on that of eq. (434) to achieve equilibrium. From the symmetry of the
B. K. D. Gairola
284
Ch.4
problem it follows that the additional displacement u should be radial, i.e. the only nonvanishing component of u is UI" = w~. In cylindrical coordinates r, cp, Z, we have
(j*Z2
= -r1Z ( p*
(441)
(j*33
=
+
p*
. v* 2p~
-
r
dV*) + !/J*1 + 2!/J*2 + 2!/J-_r 2 dr
0rv; 0OJ,
dv*/dr 2
=
0
I"
fo
0
e:" =
BZ
B2
+ 2!/J2 [ q(r) + 4n 2 r 2
0v; Ir3 O. OJ
fdv*/dr -
2
0
0
I"
0
0
0
".'3 ~ _ (442)
Substituting eqs. (435) and (442) in eq. (425) we obtain the following nonvanishing components of {/k}*:
}* =
I {1 1
2
v;,
d dr 2
1 2}* -- rdv; * & - vI"' {2 1 dv;
v; (443)
The other starred quantities can be calculated similarly in a straightforward way. We have 1* = 111* = 2q(r),
v;
where
(445)
I" ,
r
o
0
r
q(r) =
B dv; - - 22 -d q(r) ttr
r
B 2 dv*
+ -n4 2 2 -dl" r r
(446)
d v* _I"
n r
-
.1, ) ( ) o/z q r
B ¢ 6 dv; +2 z 2 -d ' n r r
(447.3)
v* = p* - 2p dv* + !/J * + ( 2 + -B2r2 -) !/J * + 2!/J ---!.-, _I"
dr
(448.5)
v* + -L.
(449)
r
dr
+ 21>4 + 4¢s, + 2¢ s + 4¢ 6 .
(450)
(451) where we have put
1
w: = u.,
Al + A z + 2A3 ,
a
=
b
= p + !/J1 + 2!/J2, B2 ¢6 B4 = -n4 2 2 (2A3 + !/JZ) + - 8 4 4' r n r
c, ~ 7:,ca -
b) -
4::7 [2A 3- VJ2 2
r:r (2A, - VJ2)]
B ( d¢6 ) + 8n4r4 r dr - 3¢6 '
(447.1)
2
(j* 11
0,
Substitution of the above relations in eq. yields three equations, two of which are automatically satisfied. The remaining equation is
C3~ 7:7 (VJ, - p -
2
3 -
(448.4)
4
r
(447.2)
0/2
~;}
(448.3)
= 2¢ 1 + 4¢ 2 + 2¢3'
2
) ( ) dv; ' p * = (A l+pqr +B-2¢222 -d
.1,* _ (A
=
(j*13
A z = 2¢1 A 3 = 2¢2
C1
o
=
B (VJI' + VJ{ + 2VJ2
dV*] d; ,
(444)
B 2 dv* 11* = 4q(r) + 2--z2 -d tt r
(j*12
A.
~~-?'
(448.2)
l
+ 2!/J*Z + - (!/J*1 + !/J*) 4n2r2 2
!/J*1
All other {/k}(g) vanish. Therefore eq. (422) may be put in the form g:l
285
Nonlinear elastic problems
§3.2
4n
Z
Z
r
(448.1)
C4
db = dr
B2
+ 4n
2r2
a)
+
4:':2 [2A3- VJ, - 7:,cA3- VJ2)}
d!/Jz B2 dr - 4n Zr3 !/Jz·
(452)
Equation (451) has to be supplemented by appropriate boundary conditions which have been discussed in the previous sections. The solution of this boundary-value
286
B. K. D. Gairola
Ch.4
Nonlinear elastic problems
§3.2
problem, which can always be obtained by numerical computations, is valid for every energy function W(1, II, III). Let us now consider the case when W is approximated by the expansion of eq. (303) in which terms up to third order only are retained. From the relations of eqs. (127) and (418) it follows that
IE = 1(1 - 3),
+ (1 -
3)].
(453)
Therefore, we may write
w=
tV,) (1 -
(/1 + -
±(V2
l28n 4r4 (5v 1
+
l8v 2
+ 16v3 ) ,
B2 A 3 = A + 2/l - 8n 2r2 (A - 3v 1
lIE = ~{(II - 3) - 2(1 - 3)J, IIIE = iUIII - 1) - (II - 3)
+
B4
3) + !(A + 2/1 + 4v2 + 8v
+ 2v3 )(I - 3)(11 -
3) - !CU
3)(1
+ v3 )(II -
B2
A4 = -4 2 3 (A tt r
+u+
-
gV 2 -
8v3 )
B4
v2
287
-
+ v3 ) + 32tt 4 rs (v 1 +
3B 4 128n4r4 (v 1 3V 2
+ 2v3 ) ·
+ 2v 2 ) , (457)
Equation (456) can be solved analytically for two limiting cases. Sufficiently far from the dislocation core where r is large compared with B/2n we get 3)2
2u
d r dr 2
3) (454)
+ ~ dUr
2
_
r dr
ur = B (A + /l + v2 + v3 ) . r2 4n 2 r3 (A + 2/l)
(458)
This equation is of exactly the same form as eq. (346) and therefore we again obtain the same solution as in subsect. 3.2.1.2. For the other limiting case, namely, when r is small compared with B/2n, eq. (456) reduces to
Using eqs. (416) and (431) we now obtain
(v 1
d 2u
1 dzz,
r + 2v2 ) - d 2 + (5V 1 + 18v2 + 16v3 ) --d r r r
(459)
which has been solved by Seeger and Wesolowski. However, we think that the situation to which this applies is better treated by atomic theory.
3.3. Dimensional changes in crystals caused by dislocations
cPs
= - !(v 2
cP1
=
cP2
=
+ 2v3 ) ,
cP3
=
cP6
= o.
(455)
In view of these relations, eq. (451) takes the form
3.3.1. General anisotropic case A theorem of Albenga [89-91J states that the average stresses in any body, which is held in equilibrium without external loads, are zero. This can be proved easily in the following way. Let v denote the region occupied by the deformed body and s denote its external surface. Since no external loads are acting on the body we have the equilibrium condition
Vi'r . (J = 0 (456)
(460)
and the boundary condition n-
where
throughout v
(J
= 0
on s.
(461)
Therefore, using the divergence theorem, we obtain
1 1 adv =
[a
+
("y'a)r] dv
~
1
"y·ardv
=
1
n'ards = O.
(462)
288
B. K. D. Gairola
Ch.4
In a dislocated crystal the stress field has discontinuities on the barrier (or dislocation surface) but the stress vector is continuous across it. Let s' denote the barrier surface and 0-(1),0-(2) be the limiting values of the stress 0 at a point of the barrier as we approach it from two sides. Then, since the stress vector is continuous across the barrier we have
ons'.
(463)
' or dv =
\7 y
v
v
I n·o-rds + I n'[(0(2) Js Js'
o(l»)rJ ds = O.
Iv JudV
0,
=
(465)
which follows directly from eq. (153). In the second order approximation we have from eqs. (132) and (314) ~ 2 +1 +-2 1 ~2 (466) J = 1 + 8 tr U(1) v R + 8 [tr U(2)\7R + I(tr U(1)\7R) - I tr(U(1) v R) J, and from eq. (318)
=
a
+
8 0(1)
8
2
=
80(1)
+
8
2
[tr
U(1)VR
+
8,
0(2)].
+
8
2 -
£(2) ~ £-
~
V R U,
8 f(1) ~ f,
(473)
+
0
+ uVr , ( C ( 3 ) " E)
f)·Vyu
(C(2)"
+ !(C(3)" f)·· fJ} dV.
In view of the symmetry properties of the elastic constants eq. (474) in Cartesian components as
i
C(2) GUm klmn GZ V n
~
dV = -
2
and
C(3)
we can write
i
{C(2) GU p GU p klmn GZ GZ V m n
GU + 2 (C(2) plmn;\ + uZ k
p
+
C(2)
(474)
C(2) GU 1) OUm kpmn ;\ ;\ uZ p uZ n
OUm OU p}
(3)
Cklmnpq ~;;-- d V. oz;
(475)
UZ q
Since C(2) is the elastic constant of the linear elasticity theory, it has an inverse the elastic compliance, such that
2 sH~n C~;;pq = bkp b1q + bkq b1p .
5(2)'
(476)
If we now multiply both sides of eq. (475) by Sr~JI we obtain the result
I Jv
OUr OZr
dV = _
~
2
I Jv
S(2) {C(2) OU p GU p rrkl klmn GZ GZ m n GUk
(2)
+
2 ( C p1m n - ; -
+
(3) GUm GU p} Cklmnpq --;Zq
UZ p
oz;
+
(2)
OU1) OUm
C k pm n -0 ~ Zp
oz;
(477)
-0 d V.
On the other hand the total change in volume undergone by the body as a result of the introduction of dislocations is given by (469) v - V =
(470)
and in the terms enclosed by square brackets we can put 8V R U(1)
£-
Substitution of this expression in eq. (465) leads to the relation
(468)
In the first term on the right-hand side of eq. (469) we can put £(1)
1 ~ + IC(2)" D + UV R·(C(2)·· £) 1 -+ (C(2)" £)·\7R U + I(C(3)" £) .. £.
C(2)"
we obtain
Substituting eq. (319) with eq. (321) in the above equation we obtain 2 2 1 +-fo = C(2)" (8 £(1) + 8 £(2») + 8 [IC(2)" D(1) + U(1)V R.(C(2)" £(1») 1 + (C(2)" £(1») . V R U(1) + I(C(3)" £(1»)" £(1)].
8
=
(467)
0(2)'
Hence, retaining terms only up to second order in fa
fa
(464)
From this result it follows that in the linear theory, since the strains are linear functions of stresses, the average strains vanish also, and in particular, there is no overall change of volume. Experimental evidence, however, contradicts this conclusion [1-3]. It was first demonstrated by Zener [92J without the explicit use of second order elasticity that the increase in the volume of the material produced by internal strains is proportional to the strain energy density. His theory was applied to the case of dislocations by Keyes [93J, Seeger and Haasen [94J and Seeger [95]. Using the nonlinear theory of elasticity Seeger and Mann [18J and Wang [96J have derived Zener's formula for isotropic materials. The most general form was derived by Toupin and Rivlin [97J, who also gave explicit formulae for cubic crystals. We proceed by rewriting eq. (464) in the form
289
and thus we can write
Iv C(2)" EdV ~ - Iv {tC(2)"
Thus we again have
Jodv = J
Nonlinear elastic problems
§3.3
(478)
1) dV.
Substituting eq. (466) in eq. (478) and using arguments similar to those leading to eqs. (470) and (471) we get
(471) (472)
Iv (J -
v - V =
Iv tr Vru dV + ~ f[(Ir uVr? -
Ir(U(l)Vr?J dV,
(479)
Ch.4
B. K. D. Gairola
290
r aUk dV + ~ f [(OUOZkk)2 _ aUk OU I] dV. oZz OZk
Jv aZk
2
(490)
(480)
(491)
By introducing eq. (477) in eq. (480) we obtain v - V =
f
Fklmn OUk OUm d V, ~ ~ UZI oz;
2
(481)
WD
="'i1 K
where F kZmn = -tSr~;iC;~ln
+
W s = f.l
s.; +
Cz~~n bkP C;;~n bkq + CH~npq) + t(b kZbmn - bkn bzm)'
(482)
It may be noted that in deriving this result we have actually used the first order approximation for the displacement field. 3.3 .2. Zener's formula
We have so far assumed that the material has an arbitrary symmetry. We now consider the simple case of isotropic materials for which C~?~n
= A bkl bmn + f.l(b km bin + bkn b1m), (483) C~f~npq = V1 bkl bmn bpq + V2[bkl(bmp bnq + bmq bnp) + bmn(b kp b zq + bkq b1p)
+ bpibkm bin + s.; b1m)J + V3[bkm(bzp bnq + b1q bnp) + b1n(b kp bmq + bkq bmp) + bkn(b zp bmq + b1q bmp) + b1m(b kp bnq + bkq bnp)J, (484) and (2) _ 1 Srrkl - 3A + 2f.l bkZ.
e
F k1mn = 6K [A
(486)
where
= 2f.l - A - 3v1 - 4v 2 , B = - 3(A + 2f.l + v2) - 4v 3,
A
K
= A+
~f.l.
(487)
Therefore eq. (481) takes the form
v - V =
r (OU k)2 d V + ~ f (OU k OUk + OUk OUI) d V 6K Jv OZk 6K OZI OZI OZz OZk '
~
where
J
WD dV + Cs
aUk OUI) - ~ (OU k)2]. az z OZk 3 OZk
(493)
(494)
(495)
f
(496)
J
(497)
WDdV,
1 Ws dV. =V
In order to reduce eq. (495) to the form obtained by Zener we use the concept of the apparent elastic constants of a strained material. It may be remarked that according to a theorem of Truesdell and Noll [17J the response of a strained material to an arbitrary further infinitesimal deformation is linearly elastic if and only if the given initial strain is maintained by a state of hydrostatic stress. Otherwise the stress depends on the rotation as well as on the pure strain. This is to be expected, since ifthe material undergoes a pure rotation, with no pure strain, then the initial stress must also be rotated to maintain zero pure strain. Thus the notion of apparent elastic constants is a natural one only for the case when the initial stress is a hydrostatic pressure. Therefore, we suppose that the material is subjected to a uniform dilatation
(488)
A = (1
+
8)1.
(498)
Then
which may be rewritten as
v - V = CD
2 az z az z
dV and hence the mean dilatation may be written v - V e = -V- = CDWD + CsWs'
-s W
s; bmn + B(bkm s; + s.; bzm)J,
[~ (OU k OUk +
(492)
= dv - dV,
(485)
It follows that F k1mn in this case is given by
(OU --k ) , OZk
It can be seen that WD and Ws are first order approximations of the energy densities due to dilational strains and shear strains, respectively. This is of course due to the fact that we have used the first order approximation for the displacement field as noted before. In any volume element of the solid the dilation is given by
1 WD=V
1
291
where
or in Cartesian components
v - V =
Nonlinear elastic problems
§3.3
J
Ws dV,
(489)
V Ru=81,
(499)
f =
(500)
81,
Ch.4
B. K. D. Gairola
292
and the stress o is related to p in the form
Nonlinear elastic problems
(501) [A']
where
= -8[6K+ te(-3K+ 9v 1 + 18v 2 + 8v 3 ) ].
= (1 + 8')1.
/3
OJ
[
(513)
001
superimposed upon the uniform dilatation of eq. (498). The deformation gradient is now (503)
The total deformation gradient is given by A*
1
= •0 1 0
(502)
We now superimpose upon this stress state an infinitesimal uniform dilatation A'
293
or
u= -pi,
p
§3.3
A*
= A'· A = (1 + 8)1 + (1 + 8)/3i1i2 .
(514)
It follows that
= A'· A = (1 + 8')(1 + 8)1,
(504)
and correspondingly
VR u* = (8 + 8' + 88') I =
E.
(515)
(505)
and
u* = -p* I,
(506)
where p*
[E. *] =
= -3K(8 + 8' + 88')
- fee 2 + 288' + 28 28')(-3K+
9v 1
+
18v2
+
8v 3 ) .
(507)
Since 8' is infinitesimal we have neglected the terms of second order ine'.The apparent bulk modulus can be defined as K' (8)
= lim p 3-e:* = (8 + I)K + t8(8 + 1)( - 3K + 9v 1 + 18v 2 + 8v 3 ) . (508) 6' ->0
+
t(1 8)/3 8
ai2 =
(1
+ 8)/3[p + 8(3K + 3v 2 +
oe = 3 0V' = 3v1 + 6v2 + "3 v3
(516)
a*
+
+
8)[p
8(3K
+
ai2 the expression (517)
4v 3 )],
where we have neglected terms of second order in modulus is
B.
Hence the apparent shear
.
3v 2
+ 4v 3 ) ].
(518)
P->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<:4
-1
x V,J.ds.
(530)
If the closed contour c is sufficiently small we can replace the integrand in eq. (530) by its value at some point on the surface and take it out from under the integral sign. The remaining integral then gives simply the area f).s of the surface bounded by the contour c, and we thus obtain (531)
In the limit when c is an infinitesimal closed contour we can write
dB = a·ds
or
(532)
where
a = 04- 1
X
Vr
or
&KZ
=
(g) f}kZ
a A- -lK aa
j
k
(533)
is called the dislocation density. It can be seen immediately that (525)
a'Vr =
0
or
(9)VI&Kl
= O.
(534)
B. K. D. Gairola
296
Ch.4
An equivalent relation was first given by Nye [99J. It expresses the fact that dislocation lines cannot end within the medium. In the same way we can generalize eq. (312) to (535)
= A·dB.
db
Inserting eq. (532) in the above relation we get db = A·a·ds,
which we can put in the form (537)
db = a·ds,
where a= A'a= A.(A- 1 xVr )
(538)
is called the local dislocation density. The component form of eqs. (537) and (538) is given by (539) (540) Here we have used the relation [compare eq. (A.23)J s' x s' = (o)f. jklgl
(541)
and the fact that {/k} is symmetric in the indices) and k. It follows from eqs. (534) and (538) that the local dislocation density satisfies the relation (A- 1'a),Vr =
A- 1 ' (a 'V r ) +
(A- 1Vr )
"
a = 0
(542)
or the equivalent relation a.Vr
+ A· [A-1Vr )
..
aJ = O.
(543)
The component form of this equation can be written as (o)\7.akl
+
]
{k}jl ] = O.
1
alj[Ak 8A-. K 8x]
-
(544)
(0)
Equations (532) and (537) can be expressed in a slightly different form if we replace ds by its dual ds as shown in subsect. 2.5. We then have dB = ~ .. dS' or dB K = ~Kjk dsjk (545) and db
= a.. dS'
db1
or
= &Ijkdsjk.
(546)
The true and local dislocation densities ~ and aare related to way 1 ~K r: ,¢KI or a = "2 a :e jk -_ .1(g) 2 c. jkl ~
p..
Vv
-
a
=
1
"2 a'E
or
iV Vv
Vv
I _
jk -
.1(0)£
2
c
,
/VIm
jkm Vv
Hence using eqs. (533) and (540) we obtain ~K _l.
1)K
-
(536)
aand a in the following (547) (548)
297
Nonlinear elastic problems
§3.4
I _
ajk -
.1(0)
2
f. jkm
m _
-
1
.1 (8(A- ) \
8x j
2
_
1)K
j) 8(A8x k '
- l-K npmiP 8(A-lK - ) p _ .1Al (8(A ) k _ 8(A--l K ) j)
(0)
f.
K
ox"
-
2
K
8x j
8xk '
(549) (550)
It may be noted that both aand a are sensitive to superimposed elastic deformations. Therefore, Noll [100J introduced another definition of dislocation density which-is more convenient in certain situations. In order to obtain this definition we substitute in eq. (532) the relation of eq. (150) in which A -1 is replaced by A-i. In this way we get dB = J a·( A-1?dS = cl·dS.
(551)
The dislocation density a = Ja' (A -l)T is invariant against superimposed deformation because both dB and dS are independent of it.
3.5. Relationship between the dislocation density and the incompatibility We have seen in subsect. 2.6 that if the relaxed elements all fit together to form a continuous body the incompatibility tensor defined by eq. (199) vanishes. This is, however, not the case here. We may, therefore, expect that incompatibility is in some way related to the dislocation density. In order to derive this relation we follow a train of thought first suggested in connection with the theory of internal stresses [101,102]. One imagines that the body with internal stresses as a whole, i.e. without a wholesale dissection, would relax in a non-Euclidean space if the constraints holding it in physical (Euclidean) space were removed. Thus internal stresses can be considered as the reactions of an elastic body to the geometrical constraint which keeps it in the Euclidean space. For instance consider a thin plate between two rigid walls. It will develop internal stresses if subjected to a nonuniform heating. However, if the constraint of rigid walls were removed it would relax into an irregular curved form which is a non-Euclidean space of two dimensions. Equivalently we can say that the relaxed elements obtained by dissecting the plate can be arranged tangential to a curved space. Following the above idea we now assume that the relaxed elements of the dislocated body, which have been turned so that the lattices in them become parallel, can be arranged tangential to a more general space. Let us consider briefly the geometrical properties of this space. (For more details one should refer to the excellent book by Schouten [103].) The simplest way of doing it is to identify the generalized space X 3 locally with the tangent Euclidean space E 3 (i.e. the relaxed elements). In other words, we associate the points of any small region of X 3 with the points of a corresponding region of the Euclidean space in a one-to-one bicontinuous manner. This property of space is equivalent to assuming that its points constitute a manifold. One should, however, keep in mind that this is only a local property of the manifold.
B. K. D. Gairola
298
Ch.4
We assume that the immediate neighbourhood of each point P of X 3 can be represented by a set of coordinates x" capable of assuming all values in the neighbourhood of thex~ which are the coordinates of P. These coordinates x" which serve to represent analytically a certain part of X 3 can obviously be chosen in an infinite number of ways. We shall take them here to be the embedded coordinates. This is legitimate because we have assumed that the body relaxes as a whole in X 3 . We shall suppose that the element of distance between two neighbouring points in X 3 is given by the quadratic form dS 2 = hkl dx" dx'
(552)
where the coefficients h kl are functions of Xi subject to the condition of being continuously differentiable to a sufficiently high order. We now associate with a point P at x~ of X 3 a point p of the tangent space E 3 in which we introduce a local frame with base vectors hk subject to the condition that h k· hi
= (hkl)O'
(553)
where (hkl)O designates the value at P of the metric coefficients h kl of the non-Euclidean space X 3 • Suppose that each point Q in the neighbourhood of Pin X 3 is brought into correspondence with a point q in the neighbourhood ofpin E 3 in the following manner. If Q has the coordinates x" then q is defined by the vector
pq = [x
k
x~
-
+
qJ(2)(X
I
-
xb)]h k ,
(554)
where the functions tp are restricted to be at least of second order with respect to the variables (x k - x~). We then say that the correspondence defines a first order representation of the neighbourhood of P. The point q is said to be the image of Q in the representation, p being naturally identical with P. According to eq. (554) the point q is defined as a function of the three scalar variables x". It follows that the x k constitute a system of curvilinear coordinates in the Euclidean space E 3 in the neighbourhood of p. This system of curvilinear coordinates has the base vectors h = k
(opq) ox k
299
If the metric of the Euclidean space £3 in the coordinate system x k is defined by
dS 2 = hif) dx" dx',
(560)
then from eq. (553) it follows that for x" = x~ we have
= hk·hl = (hiT»)o'
(hkl)O
(561)
The Euclidean space £3 and the non-Euclidean space X 3 thus have the same metric coefficients for x k = x~ and are said to be tangential at this point. In a way we can say that the generalized space is built by piecing together tiny Euclidean spaces. The topology of this space is then locally Euclidean, each neighbourhood being literally a small fragment of a Euclidean space complete with a (local) reference frame. It is clear that in a generalized space finite vectors in the usual sense do not exist. (It is easy to see that a finite length of straight line cannot be drawn on a curved surface.) However, we can define vectors and tensors in this space in the following way. We associate with each point P the base vectors of the tangent Euclidean space compatible with the metric of X 3 at that point. A tensor at P is then defined to be a multilinear function of these vectors, e.g. T = -r kl hkhl • If, for each point P, the set of components -r kl is given as a function of the coordinates x" we say that we have a tensor field. Of course, the base vector hk , too, will change from point to point. This fact that the base vectors change from point to point is nothing new, since changes of the base vectors have already been considered in Euclidean space (see subsect. 2.2 and the appendix). Here, however, the change is more radical. We have here oh ox
= (h)r~ h
k j
(562)
I
jk
or (563) In view of eq. (557) we can put
(555) 0
dR = hk dx".
(556)
1 K
-AI o(A-)
ox j
K
k
(564)
and therefore (h)r l
jk
-
(h)r l
-
kj -
A-I
o(A - 1)Kk
ox j
K
_
o(A- 1)KJ.) = 2 v. I ox" {XJk'
(565)
It follows that in the presence of a continuous distribution of dislocations the con-
Using eqs. (524)-(527) we get (557)
A and A-1 can be expressed in exactly the same form as eqs. (81) and (82), i.e.
A = Ukhk, A-1 = hkfl.
-
(h)r l
jk -
at x k = x~ which coincide with the local base given initially in £3 . Thus we see that the infinitesimal vector dR in the relaxed element which is the £3 here is also given by
Hence
Nonlinear elastic problems
§3.5
(558) (559)
nection coefficients of the natural state are no longer symmetric in the lower indices. Hence just as in eq. (A.51) we can express (h)r;k in the form (h)r;k
= Gd(h) +
T jkl
+
T l jk - T k1j ,
where we have used the notation l T jkI -- .1«h)r 2 . jk
(h)r l .) k] •
(566)
B. K. D. Gairola
300
Ch.4
are identical to aj k I and hence T j k I must be components of a tensor although (h)rjk and Od(h) are not. In fact, if we apply the commutator (h)V]. (h)V m - (h)V m (h)V. to Uk we find ]
It is clear from eq. (565) that the
( h)V ] . (h)V m _
(h)n
V m
.)uk V]
(h)n
=
T jk
I
(h)R. k u l jml
+
T I n Uk jm v I .
(568)
This equation shows that T j k I are, indeed, the components of a tensor. Following Cartan [104] we refer to this tensor as the torsion tensor. In order to understand the meaning of torsion let us consider two vectors P OP1 = h k dX~l) and P OP2 = hk dX~2) drawn from a point Po at x k . A parallel transport of the vector P OP1 to the point x" + dx~2) produces the vector I P 2 P 1' -- (d X(1)
-
(h)r I jk
d X(2) j d X(1) k )h I'
(569)
Similarly, the vector P 1 P~ which is obtained by a parallel transport of Po P 2 from x" to x k + dX~l) is given by I P 1 P 2' -- (d X(2)
-
(h)r I jk
d X(1) j d X(2) k )h I'
Nonlinear elastic problems
§3.5
However, when the parallel transport is made along a closed circuit there is no change in this vector because the curvature tensor vanishes, i.e.
»n.jmk I =
d-s jk A- - 1 '91
-
-
A- - 1 db .
dB.
]n
_
mk
(h)r I (h)r,: mn]k
= 0
(574)
.
(575)
K j k I,
K j kl
+ T Ij k - T k Ij = aj kI + aIj k - ak Ij _ l.(g) m + (g) m - 2 f jkm lX I fljmr:t k =
T j kI
(g)
fkImr:t
m ) j .
(576)
Proceeding in the same way as in subsect. 2.6 we obtain the following equation in place of eq. (198): (h)R
jmkl -
(h)R* jmkl -
+
2[(g)V
2ejlpKmkn -
j
K mki -
2Kjlpemkn
(g)V
+
m
K
jkl
-
hnp(K jip
2emlpKjkn -
K mkn - K mip K
2Kmlpejkn)],
jkn
(577)
where (h)Rjmkl is the Riemannian part of (h)R j mkl given by eq. (195). Therefore instead of eq. (199) we have now
=
n*qr
+ l.(g)t:qjm(g)t:rki [(g)V.K 2 ] mkl
-
hnp(K. K ]lp mkn
where n*qr is given by eq. (199). Note that (h) R j mki is not symmetric with respect to the interchange of the indices jm and kl and hence n" is also not symmetric in q and r. The antisymmetric part of eq. (578) when multiplied by (g)f qr s yields (g)n v jIX j s
+
h np(2 e j sp - K jsp )IXj n
--
0.
(579)
In view of eq. (575) and the fact that (572)
In view of eq. (535) we see that
A- 1 · db =
(h)r~ (h)r n
where
tt"
- I IXj k
+
ax"
OX]
= Uk, lJ(h) +
(h)r j kI
(571)
In other words, a closed circuit in the form of a parallelogram in the deformed state becomes an open circuit in the natural state with a closure failure given by eq. (571). But this is just the way we define the true Burgers vector by using Frank's Burgers circuit. N ow, if we take into consideration the antisymmetry of T j k I in the indices j and k and use eqs. (565), (143) and (557) we can write eq. (571) in the form
o(h)r~ jk
o(h)r I
~
This fact can be verified in a straightforward way by substituting eq. (564) in the above equation. Although this equation resembles eq. (194) it is quite different because the connection (h)rjk is given by eq. (566) instead of eq. (193). We now put (h)r j kI in the form
(570)
In a Euclidean space the points P{ and P~ would coincide and hence the figure obtained by the above operation would be a parallelogram. In the present case, however, the figure does not close because the points P{ and P~ do not coincide because the material vectors change their direction on parallel transport. (The term material vector denotes the vector joining two material points.) What we mean to say is that the material vectors which were parallel in the Euclidean sense in the deformed state are still being looked upon as parallel in the natural state although they may not be so. It is easily seen that the closure failure is given by
301
(573)
That means that the expression (571) does, in fact, represent the true Burgers vector. Kroner [24J has demonstrated this fact in a different way by considering a Cartan displacement of the tangent £3 which corresponds to Frank's Burgers circuit. We now consider another interesting characteristic of %3' We have seen that the parallel transport of a material vector from one point to another changes its direction.
(580)
we see that eq. (579) is identical to eq. (544). The symmetric part of eq. (578) on the other hand is given by (581)
where
n" = '"-
eqr =
l.[(g)fqjm(g)frklhnp(K.
2
(582)
[(g)fqjmVjr:trmJ(qr),
it» K
mkn
-
2K. e Jip mkn
2e. K )](qr) ]Ip mkn .
B. K. D. Gairola
302
Ch.4
Here round brackets indicate symmetrization with respect to the indices enclosed by them. We can see that eq. (581) is the desired relationship between the incompatibility tensor rr* and the dislocation density a. In view ofeq. (199) it can be put in the form (Inc, e)qr = n"
+
Qqr
or
(584)
Nonlinear elastic problems
§3.6
303
(or the strains are infinitesimal) since it is insensitive to local orientation of crystallographic axes. Willis [29J on the other hand avoids this difficulty by using the relation of eq. (538) and the constitutive equation in the natural state coordinates. Before proceeding further we give below the approximate constitutive equations relating Cauchy stress and Eulerian strain for isotropic materials which were used by Kroner et al.
where Qqr
+
=
eqr
=
l.[(g)fqjm(g)frklhnp(_ 2
3.6.1. Second order constitutive equations in the deformed state coordinates for isotropic materials
~qr
2e.IIp
+ K.lip )(- 2emkn + K mkn )J(qr) .
(585)
This equation is the nonlinear generalization of the well-known relation Incre = 1]
In order to obtain equations relating the Cauchy stress to the Eulerian strain e we rewrite eq. (285) as
(586)
which was derived by Kroner [44]. Equation (584) was called the fundamental geometric equation by Kroner and Seeger. In this section we have mainly followed Kroner and Seeger because only they have applied their method to concrete problems. However, the idea that the geometry of a dislocated crystal can be appropriately represented in terms of non-Euclidean space was first introduced by Kondo [2lJ and independently by Bilby, Bullough and Smith [22J, reaching its culmination in the essay of Kroner [24]. The notion remains popular and has been further extended and developed by various authors. A fairly representative account of these developments can be found in the proceedings of two recent conferences [105, 106]. 3.6. Determination of the stress and strain fields due to a given dislocation density We now consider the problem of determining the stress field when the dislocation density is given. In principle, one can use either the true dislocation density or the local dislocation density. However, the true dislocation density is not so straightforward for some applications as it involves both the natural state and the deformed state. Moreover, in practice the experiments yield the local Burgers vectors of dislocations. Hence it is often more convenient to work with the local dislocation density. In the following we shall describe two methods of successive approximation which have been used to determine the stress and strain fields generated by a given local dislocation density. Both methods make the tacit assumption that the macroscopic stresses which are observed are not appreciably influenced by the detailed boundary conditions which should be applied at the individual dislocation cores. Otherwise the two methods are. quite different. Kroner and coworkers [23, 25J use the relation of eq. (584) to derive an inhomogeneous biharmonic equation for stress functions which are related to the Eulerian strains through an approximate constitutive equation in the deformed state coordinates. It can be seen that this method is not only elegant but also leads to simpler calculations than the displacement function method when applied to the problem of singular dislocations in isotropic media. However, the method presents difficulties when applied to anisotropic media. The Eulerian strain, as pointed out before is a satisfactory strain measure only when the body is isotropic
1 oW
oemn
-_ 1M
- J oemn o(A -l)M k A
I
_ ~ oW h - Joe In kn
(587) or (J
1 oW J oe
loW J oe
= - - h = - - (I - 2e)
(588)
'
where we have used the fact that WeE) = Wee) for isotropic materials, which follows from eqs. (291) and (125). We now proceed in the same way as in subsect. 3.2.1.1. We assume that both (J and e can be expressed as a power series in a parameter s: CIJ
e =
I
(589)
ene(n),
n=l
in which e(l) corresponds to the infinitesimal strain. We also assume that Wee) can be expanded in terms of e in the same form as eq. (286), i.e. Wee) = 1(c(2)" e) .. e
+
.g[(C(3)·· e) .. eJ .. e
+
(590)
For isotropic materials we get (see e.g. eq. (303» W(Ie' IIe, lIle)
= (lA' +
Ji')Ie2
-
2Ji'Ie
+ .g(v~ + 6v~ + 8v~)le3
- 2(v~
+ 2v~)leIIe + 4v~llle +
B. K. D. Gairola
304
Ch.4
The relationship between A', u', v~, v~, v~ and A, u, v 1, v2, V 3 can be obtained easily if we replace I, etc. by IE. etc. using eq. (125) and compare the resulting expression with the expansion W(IE.' lIE.' IIld. In this way one finds that
A' = A, v~
§3.6
o=
= t(v 1 + 18v2 + 44v 3 ) , V~
3 v2)'
V2 -
= 3u +
(592)
V3 .
qJ = Jo :
81Ie/8e = I, I - e, 8111e/8e = e
2
18W
J a; (I
+
(593.1)
The function 2.9. From
(593.2)
bqJ
(593.3)
lle - lee '
=
e-
+
4v~e2
(601)
is the enthalpy function, as we have already mentioned in subsect.
qJ
Jo>: 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 -
~
oalV. + oaW) , OZ2
~~F(l)
= 2f.1(,
=
OZl
2 02 eW _ 02eW_ 02eW= oaW _ oa~11, oZ1 0 Z2 0 z~ 0 zi 0 Z1 0 Z2
(646)
(650)
Zum
(651)
--1].
1- m
The function e~ll can be calculated in a straightforward manner from the first three of eqs. (646). Similarly the functions' and 1] are determined in a simple way by the equations
and 0'(1)
(652)
= Inc r X ( l )
reduces to (653) (654) (647) where F(l) =
X(lh3
is called Airy's stress function and ox~ll
oxW
(j~2d
a;-2 - a;-1
is called Prandtl's stress function. In view of eqs. (606), (633) and (647) we have
e(11l )
[fP F 2f.1 OZ~ 1
1
m+l
ell] = _~ 0 F(l) , 2f.1 OZl OZ2
The integrations of eqs. (646) to (648) yields (j~ld,
(AF Ll
(1)
(l))J + 0'33 '
~
(2)
=
= 2f! (O!/JW OZ 1
.=
~~F (2)
~ [~F(2) + 2f.1(m + m
_ o!/Jll1 + OZ
1) (Q' -
!/JWJ,
QII),
(655)
(656)
2
Lum [0 2 ,/,(1)
02,/A1) 02.1,(1) _'P_l_l + _'P_2_2 _ 2 _'P_l_2 + Q(l) 1- m OZ~ ozi OZ1 0Z2 33
--
2
+
~ (111//l'j + Qi'l + Q~D}
(657)
Ch.4
B. K. D. Gairola
312
I1I1F = 2mf.1 [m + 1 A' 11 (2) m - 1 m 2
/I
82Q'/8zi = -Qi1d; 82 Q' /OZl 8z 2 = QW
82Q'/OZ~
= -QW, (658)
and -Q~l,
(659)
3.6.2.1. Example. Let us now proceed to the concrete example of a continuous distribution of screw dislocations which are parallel to the Z3 direction. That means the only nonvanishing component of the dislocation density is 0:3 3 (z l' Z2)' It can be seen that in this case 11 vanishes and therefore F(l)' e~ll and aWare also equal to zero. Thus the only first order equation we are left with is
I1cP(l) = - f.1o:W
+
const.
(660)
It follows from eq. (648) that the only nonvanishing components of e(l) are ei l and 1
e~l]. In view of eqs. (628) and (647) the components of "'(1) are given by
l/JW
=
_A~[(8cP(l»)2 + (8 cP(l»)2J _ A~(8cP(l»)2, 8z 1
8z 2
313
{(8cP(l»)2 + (8 cP(l»)2} 8z 1. . 8z2 2 cP(l»)2 2 cP(1»)2 2 + A~ { 82(8 + 82(8 _ 2 8 (8 cP(1) 8 cP(1»)} 8z 1 8z 2 8z 2 8z 1 OZ1 8z2 8z 1 8z 2 2 2 2 + ~ {8 cPy) 8 cPi1) _ ( 8 cP(l) )2}J' (666) f.1 8z 8z 8z
where Q and Q are determined by the equations I
Nonlinear elastic problems
§3.6
1
18z2 where Q' satisfies eqs. (658) with QW = Qi 1d = Q~ld = 2
o.
A singular screw dislocation along the z3 axis can be considered as a special case of the above example when the dislocation density has the form (667)
where b( z 1) and b(z 2) are delta functions. In subsect. 3.2.1.2 we considered a singular screw dislocation along the z axis of an infinite hollow cylinder with inner radius r c and outer radius reo Let us consider this case again. For the sake of convenience we now use cylindrical coordinates r, tp, and z and hence we write 0:3 3
= b b(r)
0: 3 3
= 80:~~ = -
or
8z 1
./A1) _ 'fJ12 -
8b(1)
(668)
b(r).
The first order solution in this case is well known [91]. We have ./A1) _ 'fJ22 -
,/-.
'Wi ~ -A; [(iJiJ~:)r + eiJ~~))] l/JW
(661)
= !/Jill = O.
-
-
4
8
8 8
e(l) e(l») 8 e(l) e(l) ~~-~~. ( 8z 8Z 8z 1 8z2 2 1
(662)
82cP(l) _ (8 2cP(1) )2J. 8z~
8z 18z2
11 cP(2) = const,
J
+
2fl(m
+
I)
{Q' + A~ eiJ~~'Y + A~ eiJ~~)n }
(669)
nr,
= (IncrX(l»)
f.1b 2nr
(670)
Therefore we can proceed directly to the second order equations (664)-(666). First of all, we note that cP(2) = 0 since there are no external forces acting on the body. It follows that
= 0.
(671)
Substituting eq. (669) in eq. (666) and using eqs. (608) and (609) we finally obtain (663)
I1I1F(2) =
Accordingly, eqs. (655) and (657) now give
cr~i = ~ [M(2
a~)11
"",(2) V¢z
Substituting eq. (648) in eq. (662) we get
Q(l) = _~[82cP(1) 33 f.12 8zi
=~
and accordingly
Furthermore, we find that all Q~Jl vanish except Q (331 )
f.1b(l) 1
'fJ(1)
4N
7 =
1
2
zN 1111 (In r) ,
(672)
where (664) (673) (665)
Ch.4
B. K. D. Gairola
314
Obviously the general solution of the inhomogeneous equation (672) for an infinite medium is (674) where f is a biharmonic function satisfying the homogeneous equation fJ.fJ.f = O. The function f should have the same rotational symmetry as the first term on the right-hand side of eq. (674). Therefore we put
Nonlinear elastic problems
§3.6
and hence the mean value of fJ. (2)F is zero. Similarly from Albenga's theorem (see subsect. 2.3) it follows that the mean of O'zz = 8 2 O'~;) is also zero. Therefore taking the mean of eq. (664) we obtain
Q'=
O'rr
-
=
(Inc r X (2 ) ) rr N
2" In r
r
+
_
-
2d1
0' (2) -_ zz
(btl)8nv~ + N ) [ -r1 -2
2
2
r; -
r~
reJ
In- .
(685)
rc
Finally we obtain the stress field o by putting
~ 8 F(2) ~ 8F(2) r2 8 qJ2 + r 8r 2
(2) _
(684)
Thus we have
(675) where d 1 and d 2 are constants. Hence, using the formulae in appendix A.3, we obtain
315
_ U -
d2
+ 2' r
(676) (677)
B u(l)
+
2
and
B u(2)'
b
2
=
8
2b2
(1)'
The result is
r
r 2
r)
o = N' r2 In - + r2 In ~ - r2 In ~ 9 9 e r~ c r r r r c
(
(678) In view of the boundary conditions at
r
= rc and
r
(686)
=
r,
where
it follows that
2
(679)
, b N =8 tt 2 r 2 (r;2 - r 2) c Nil
(680) Thus we get 0'(2)
=
rr
0'
(2) ({UP
N
r2 (r; -
= -N2 [ 1 r
r~)
( r - r2 In ~ r) , r2 r In - + r2 In ~ c e rc r rc I
r; -
r~
( r_2 In -r
re
(681)
re )
In - + r In - . ere + re r rc 2
2
(682)
In order to calculate the remaining component O'~;) we proceed in the following way. We first note that fJ. F(2) -_
(2)
0' rr
+
(2) _
0' ({)(P -
N [ 2"I -
r
reJ
2 2 In2 r; - r; r c
(683)
= N' +
[2/-1---1 m - 2(, ,)J m V2+V 3
bi v: 3
8n r (r; - r~) 2
2
'
(687) (688)
It can be easily verified that the solution obtained here agrees with that obtained earlier by the displacement-function method. Kroner et al. have also considered edge dislocations. For a continuous distribution of edge dislocations which are parallel to the Z3 direction the only nonvanishing components of the dislocation density are 0:3 1 (z, z) and 0:3 2 (z, z). In the absence of external forces it follows from eqs. (646), (652), (653), (650) and (648) that ( as well as e\li, e~li and e~li vanish. One also finds that apart from Q33 all other components of Q vanish, and that Q 3 3 is a function of the derivatives of F alone. Thus, in this case, only Airy's stress function enters the first order equations. On the other hand in the case of screw dislocations, as we have seen, the first order equations contained only Prandtl's stress function. However, in general both Airy's and Prandtl's stress functions enter the second order equations. Kroner et al. have solved these equations for the case of a singular edge dislocation. We shall not give the tedious details of
Ch.4
B. K. D. Gairola
316
their calculations. Their solution in itself is quite lengthy, but it takes a much simpler form at a distance large compared to rc ' This form of the solution reads (J
rr
(J
rqJ
sinq> + M2 5[ (2M 1 + M 2) In-r + 41M 2 cos 2q> = Mor r rc cosq> = - M -
r
0
+
M2
c
2r 2
317
CIJ
I
f3 =
8
n
(692)
f3(n)'
n=l
Using eq. (528) we obtain -
+
A = I
8f3(1)
+
2
8
(f3(1)' f3(1)
+
+
13(2)
(693)
c
(r)
sinq> M 5[ 1 ] = Mor + -r2 (2M 1 + M 1 ) 1 - In -rc - 4M2 cos 2q> .'
Substitution of eqs. (691) and (693) in eq. (538) yields
(Jrz
=
It follows from the assumption of eq. (619) that
(Jzz
sin q> = 2vMo - +
(J
qJqJ
(JqJz
r
M5 {v(2M
-2
r
1
-813(1) x
VI' = VI' = -
13(1) x
+ M1 )
13(2) x
~J}'
(689)
t=7 VI' -
1
8
+-
[13(1)'(13(1) x VI')
+
+-
13(2) x VrJ
+
(694)
(695)
a(1)' 13(1)' (13(1) x
VI') =
(696)
13(1)' a(l)' etc.
Furthermore we have
t(A T . A - I) = t[8(f3(1) + f3il) +
f =
where
flb M o = 2n(1 - v)' 2fl (1 - v)
M1
+
v(l
+
[(1 - v)8fl2(1 v)A~
~[~ 1 - v 8fl
M3 =
~ [4v(1 4fl
+ (1
(I
+
+
+ v)(1 +
(1
V
2
+
V)3 A'
J- 1
1
(1 - 2v) (v~
+
+ 2v~) -
f3il)'f3i1)
+
f3i1)'f3(l)
+ 13(2) +
f3i2)
+ ... J,
-1)
= 1-
81{3(1)
+ ....
(698)
CIJ
_ a -
'\'
n
L, E
(699)
a(n)'
n=l
The nth term a(n) may be put in the form
(v~ + 6v~ + 8v~)
(700) where C1~) is a function of 13(1)' 13(2)' ... , f3(n-l) only, i.e.
2v~)J,
2v3 J,
a~) = C1~)(f3(1)' 13(2)" .. , f3(n -
(690)
(691)
(701)
1)'
For instance, in the second order approximation we have a(1)
and A~, A~, A~, A~ are given by eqs. (592) and (608), respectively.
I - f3.
+
Inserting these relations and eq. (286) in eq. (274) we obtain a series expansion for a:
3.6.3. Willis's method This method utilizes eq. (538) relating the local dislocation density with the distortions, the equation of equilibrium (614) and the constitutive equation (274). It will be seen that this method of successive approximation with minor modifications is the same as that ofSignorini discussed in subsect. 3.2.1, with the difference, however, that the distortions are not expressed in terms of a displacement field. Instead we write
A -1 =
= [det (A)J-l = det (A
)A ;}
v)A~ - VA~J'
-2(1 - 2v) (4v - 3v 2 - 1) (v~
v~, v~, v~
2 (f3 (1)' f3 (1)
and from eq. (131)
2v)
- v) (v~ - fl) - !(1 - 2V)3
~ [8flV + 2fl
8
(697)
M2 =
M4 =
=
a
= 0,
-2M3 + cos 2'1' [(2M, + M.) - vM2 In
and
Nonlinear elastic problems
We now assume that
(1 - 41n-rr ) ] ,
In -r) , r
M5sm. 2q> ( 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""<1'1'''"......
(A.20)
Ch.4
B. K. D. Gairola
328
indices). These quantities are, therefore, components of the unit tensor I which may be expressed in the following way I = GKL G K G L = (jKL GK G L = G KL GK G L.
(A.31)
In fact, fGlEJKL and (GlEJKL are also components of an anti-symmetric tensor € as can be seen by writing - e
= I x 1=
GKGK
X
GLG L
=
GKG K
GLGL.
X
(A.32.2) 2. Covariant differentiation Let u be a vector localized at some point P(X l , X 2 , X 3 ) . If at every point of some region V about P we have a uniquely defined vector u, we refer to the totality of vectors u in Vas a vector field. We suppose that the components of u are continuously differentiable functions of the coordinates X K so that u has the representation u(Xt, X
2
,
X
3
)
=
UK(Xt, X
2
,
X
3
)
GK(Xt, X
2
,
X
3
).
du
=
GK d U
K
+
U
K
dGK
=
(aUK GK a x J
+
U
J K aGK) ax J d X .
(A. 34)
For every J and K the derivative a GKlax J is a vector, and as such it may be represented as a linear combination of the G's. Denote the coefficients of this linear combination by (Glr YK' so that aGK_(GlrL G ax J JK L'
-(G)r~L U L
=
(j UK
d XJ.
(A.37)
Therefore in curvilinear coordinates, the difference in the components of the two vectors after translating one of them to the point where the other is located will not coincide with their difference d UK before the translation. Thus in order to compare u(X\ X 2 , X 3 ) and u(X l + dX\ X 2 + dX 2 , X 3 + dX 3 ) we subject the former to an infinitesimal parallel displacement to the point X K + dX K. Then its components UK change to UK + bU K. The components of u(X l + dX 1 , X 2 + dX 2 , X 3 + dX 3 ) , on the other hand, are UK + d UK. It follows that the difference between the two vectors which are now located at the same point is given by du
(A.33)
We now consider the vector change du in u as the point P(X l , X 2 , X 3 } assumes a different position P(X l + ss>, X 2 + dX 2 , X 3 + dX 3 ) . From eq. (A.33) it follows that
329
of a vector parallel to itself. Thus we can say that under a parallel transport of a vector its components in Cartesian coordinates do not change. If, on the other hand, we use curvilinear coordinates, then in general the components of a vector will change under such a transport. We can see that this change is due to the change in base vectors and we can express it in the form
(A.32.1)
Using the relations (A.20) and (A.23) we can put the above equation in the form
Nonlinear elastic problems
§A.2
=
(dU K - bUK)G K
=
(aUK axJ
+
(Glr K UL)dXJG
JL
K
or au _ (aUK axJ axJ
+
(Gl K
rJLu
L)
GK·
+
(G)r~L U L d XJ) GK.
(GlV UK J
.
=
aX! + (GlrK
a UK
UL
(AAO)
JL'
(A.35) Thus we can write
(A.36)
To understand the meaning of this equation let us look at eq. (A.33). The vector du is actually the difference of vectors located at different infinitesimally separated points X K and X K + dX K. However, for the two vectors to be subtracted from each other it is necessary that they be located at the same point in space. In other words, wemust somehow transport one ofthe vectors to the point where the second is located, after which we determine the difference of two vectors, which now refer to one and the same point in space. In Cartesian coordinates d UK is just the difference of the com..,.u.,,, of two infinitesimally separated vectors. This means that when we use coordinates the components of the.vectora should not change as a result of operation. But such a transportation is precisely the transportation tJ'U'....
(A 39)
The quantities enclosed by the brackets on the right-hand side of eq. (A.39) may be considered as the components of aufax J. Since these components of the differential aula x J play an important part they deserve a name and notation. They are called covariant derivatives of the components UK and they are denoted by (for example):
F''s are called connections with respect to the G's. If the relations of eq. (A.35) are inserted into eq. (A. 34) one obtains du = (d UK
(A.38)
(AAI) which shows that (GlVJ UK are the components of a tensor of second fact that GK·G L
=
the
(jK L
we can easily derive aG
a
K
=
_(Glr K G L
JL
and (Glv U
J K
=
aUK - (Glr L U axJ JK L'
(A.42)
o,
1\... i».
srairota
Ch.4
With the above procedure we can obtain covariant derivatives of the components of a vector or tensor provided the (Glr's are known. However, if the GKL is known then (Glr can be found. We differentiate eq. (A. 14) and use eq. (A.35) to obtain aG KL _ aGK G axJ - axJ' L
+
aG L GK' a x J
(A.44)
i.e. (GlVJG KL = O.
= (GlVJfKLM = O.
(A.46)
Equation (A.44) establishes the relation between the G'« and (Glr's. They must be solved for (Glr's. In order to do this we interchange J and K and then J and L in eq. (A.44) to obtain
+
aGJL axK
= (GlrM G
JK aG --L ax
= (GlrM LK G JM + (GlrAfJ GKM.
LM
(GlrM G KL
(A.47)
JM
and
+
aGJK _ aGKL ax L oX!
= ((GlrPfL +
(A.53)
(GlrN = IG JN (aG JL KL
KJ
JL
+
aG JK _ aG KL) ax L axJ
{NKL}(Gl ,
(A.54)
where [KL, JJ(Gl and {~L}(Gl are called Christoffel symbols of the first and second kind respectively. It is worth noting that (Glr~L are not the components of a tensor; in fact if they were, the value of this tensor would be zero in Cartesian coordinates since the base vectors t, are constants in this system. This is a contradiction, since a tensor is independent of its representations. One can similarly determine the covariant derivatives of the components of a tensor of arbitrary rank. To obtain the covariant derivative of the components A::: with respect to the base vectors GK , GK we add to the ordinary derivative aA:: :/ax J for each covariant index K (e.g. A:KJ a term - (GlTYKA:i: and for each contravariant index K (e.g. A:~:) a term +(GlrfLA:~:- For instance, J
L
axJ
+
(GlrK "[M JM
L
_ (GlrM 1;K JL M'
(A.55)
Note, however, that if a tensor is expressed in terms of components with respect to base vectors of different coordinate systems we should modify the notation accordingly. Consider, for example, the tensor function A in subsect. 2.2 which has the components A k K with respect to the coordinate systems x k and X K . In this case we have
+ ((Glr M _ (GlrM ) + ((GlrM _ (GlrM ) JK
ax K
2
= G IN [KL, JJ(Gl =
(A.48)
LJ
(Glr~K)'
(A.49)
We add ceGlr~L - (GlrAfK) on both sides and use the relation GJNGJM
(Glr;K = (GlriJ'
(GlV "[K = a1;K L
If eq. (A.44) is subtracted from the sum of the last two equations, one gets aGJL axK
Since R is a unique function of xl, X 2 , X 3 , it follows that the (Glr's are symmetric with respect to their lower indices, i.e.
(A.45)
This equation implies that the lengths of vectors are conserved under parallel transport. From this equation it also follows that
KJ
j5l
Therefore eq. (A.5l) reduces to
= (GlrAfKGLM + (GlrAfL GKM,
(GlV/GlfKLM
Nonttnear etasttc prootems
~A.2
= b NM
(A.50)
aA
axJ
a
k
K
= axJ (A K9kG ) _ aA\ K - ax J 9kG
+
k (a9 k j K A K ax jA JG
ecK)
+ 9k ax J
and thus obtain k
(GlrN = KL
IGJN(aG JL
axK
2
+
aA K + = __
aG JK_ aGKL)+l((GlrN _ (GlrN ) axL ax J 2 KL LK
_ (GlrM ) G JNG + l((GlrM 2 JK KJ LM
M _ + l((Glr 2 JL
( ax J
_ (Glr L A k ) 9 G K JK L k
= (VJA\)gk G K
(GlrM )G JNG LJ
(glr~ Al Aj ]1 K J
KM'
(A.56)
where VJA kK is called the total covariant derivative of A k K'
(A.5l) However, from eqs. (A.35) and (A.12) we get
aG
L
= aXKJ ' G =
aR 2
~
__
T
TT'
G
3. Physical components
L
(A.52)
So far we have formulated the equations in general curvilinear coordinates. I-I0vvever, when we deal with concrete problems we very often use orthogonal curvilinear
D. ft. V. VUtfUIU
Ln. 4
coordinates. In such cases it is more convenient to use physical components of tensor quantities. The physical component of a tensor at a point P relative to a system of orthogonal curvilinear coordinates are simply the Cartesian components in a local set of Cartesian axes, tangent to the coordinate curves through P. The unit base vectors G(K) of the local Cartesian frame are defined by GK
GK
= ;r;-;:;- = r;:;---'
G(K)
V
V
G K· G K
(A.57)
=
8G(K) _ ax -
From eq. (A.61) it follows that
r. (J)(K)(L) --
(A.58)
G(KK)'
au a
=
x
K
= -
P(K)G
az
M • --a K'M P(K) X
I
(A.60)
G K·
=
G(K)
and the base vectors i K of the fixed
• Q(K)M'M'
(A.61)
Hence if T L 1 ••. t.; denote the Cartesian components of a tensor T in the fixed Cartesian system, while the physical components in the orthogonal curvilinear system are T(K 1) ... (K n ) , then T(KJ) ... (K n )
=
Q(KJ)Ll ... Q(Kn)Ln T L 1 ... Ln'
(A.62)
Note that the covariant and the contravariant physical components are the same because the base vectors G(K) are orthogonal unit vectors. It is also obvious that Q(K)MQM(L)
a
K G
aXK
=
=
a G(K) ax
Gl
aX(K)
=
a
P(K) ax
+
r(J)(L)(K) VeL)
)
K'
(A.68)
G(K) V(J) V(K) .
'
(A.64)
G2 = - R sin ¢il
+ R cos cjJi2 , (A.70)
G 3 = i3 ·
Therefore we have
GK L
1 0 0] = 0 R 0 , [o 0 1
KL
2
G
=
[~
o 1/R 2
o
D
(A.71)
It follows that G(l)
=
GR
= cos ¢il + sin ¢i 2 ,
(A.72.1)
G(2)
=
G4>
= - 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
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Nonttnear etastic problems
j41
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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
