Messerschmidt U. Dislocation Dynamics During Plastic Deformation
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Contents XI
5.2.3 Experimental Determination
oftheStressComponents ..........................187
5.2.4 SteadyStateDeformation ..........................192
5.3 Plastic Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
Part II Dislocation Motion in Particular Materials
6 Semiconductors ............................................207
6.1 CrystalStructureandSlipGeometry.......................207
6.2 MicroscopicObservations.................................209
6.3 DislocationDynamics....................................213
6.4 Recombination-enhanced Dislocation Mobility . . . . . . . . . . . . . . . 217
6.5 MacroscopicDeformationProperties.......................218
6.6 Summary...............................................220
7 Ceramic Single Crystals ...................................221
7.1 Alkali Halides...........................................221
7.1.1 Crystal Structureand Slip Geometry ................222
7.1.2 DislocationDynamics..............................223
7.1.3 MacroscopicDeformationProperties.................223
7.1.4 Summary.........................................229
7.2 MagnesiumOxide .......................................231
7.2.1 MicroscopicObservations...........................231
7.2.2 Statistics of Overcoming Localized Obstacles . . . . . . . . . . 234
7.2.3 Kinematics of Overcoming Localized Obstacles . . . . . . . . 239
7.2.4 DislocationDynamics..............................242
7.2.5 Macroscopic Deformation Properties and Discussion . . . 243
7.2.6 Dislocations in the Plastic Zone of a Crack . . . . . . . . . . . 249
7.2.7 Summary.........................................252
7.3 ZirconiaSingleCrystals ..................................253
7.3.1 Crystal Structure and Slip Geometry
of ZrO
2
–Y
2
O
3
alloys...............................253
7.3.2 Microscopic Observations in Cubic ZrO
2
.............254
7.3.3 Dislocation Dynamics in Cubic ZrO
2
.................260
7.3.4 Macroscopic Deformation Properties of Cubic ZrO
2
....261
7.3.5 DeformationMechanisms...........................263
7.3.6 Summary of Cubic ZrO
2
...........................273
7.3.7 Tetragonal ZrO
2
..................................273
8 Metallic Alloys ............................................281
8.1 PrecipitationHardened Aluminium Alloys ..................281
8.1.1 Al–Zn–Mg........................................282
8.1.2 Al–Ag ...........................................285
8.1.3 Al–Li ............................................287
8.1.4 Summary.........................................293
XII Contents
8.2 DislocationGenerationinMetals..........................293
8.3 Oxide Dispersion StrengthenedMaterials...................297
8.3.1 Microscopic Observations
inOxideDispersionStrengthenedAlloys .............298
8.3.2 MacroscopicDeformationProperties.................303
8.3.3 DeformationMechanisms...........................304
8.3.4 Summary.........................................308
8.4 Plastic Deformation During Fracture
of Al
2
O
3
/NbSandwichSpecimens .........................309
9 Intermetallic Alloys ........................................313
9.1 Introduction ............................................313
9.2 Ni
3
Al ..................................................316
9.2.1 Microscopic Observations and Dislocation Dynamics . . . 316
9.2.2 Modelsofthe FlowStressAnomaly..................319
9.3 γ-TiAl .................................................322
9.3.1 Crystal Structureand Slip Geometry ................323
9.3.2 MicroscopicObservations...........................324
9.3.3 Macroscopic Deformation Parameters . . . . . . . . . . . . . . . . 331
9.3.4 DeformationMechanisms...........................333
9.3.5 Summary.........................................340
9.4 NiAl...................................................340
9.4.1 Crystal Structureand Slip Geometry ................340
9.4.2 MicroscopicObservations...........................341
9.4.3 Macroscopic Deformation Parameters . . . . . . . . . . . . . . . . 346
9.4.4 DeformationMechanisms...........................348
9.4.5 Summary.........................................355
9.5 FeAl...................................................355
9.5.1 MicroscopicObservations...........................355
9.5.2 Macroscopic Deformation Parameters . . . . . . . . . . . . . . . . 360
9.5.3 DeformationMechanisms...........................362
9.5.4 Summary.........................................368
9.6 Molybdenum Disilicide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
9.6.1 Crystal Structureand Slip Geometry ................369
9.6.2 MicroscopicObservations...........................370
9.6.3 Macroscopic Deformation Parameters . . . . . . . . . . . . . . . . 380
9.6.4 DeformationMechanisms...........................383
9.6.5 Summary.........................................389
9.7 Conclusions on Intermetallics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
10 Quasicrystals ..............................................393
10.1 Structureof Quasicrystals ................................394
10.1.1 Quasicrystalswith IcosahedralSymmetry.............396
10.1.2 Quasicrystals with Decagonal Symmetry . . . . . . . . . . . . . 398
Contents XIII
10.2 DefectsinQuasicrystals..................................400
10.2.1 Vacancies ........................................400
10.2.2 PhasonDefects ...................................400
10.2.3 Dislocations ......................................403
10.3 MicroscopicObservationsofDislocations ...................408
10.3.1 i-Al–Pd–Mn ......................................408
10.3.2 d-Al–Ni–Co.......................................422
10.4 MacroscopicDeformationParameters ......................431
10.4.1 i-Al–Pd–Mn ......................................432
10.4.2 d-Al–Ni–Co.......................................438
10.5 Mechanisms of Dislocation Motion and Plastic Deformation . . . 440
10.5.1 Glide orClimbMotionofDislocations ...............441
10.5.2 Componentsof theFlowStress......................443
10.5.3 FormationofPhasonFaults.........................443
10.5.4 Long-RangeDislocationInteractions.................445
10.5.5 Activation Parameters of Plastic Deformation . . . . . . . . . 446
10.5.6 FrictionMechanismsofDislocationMotion ...........448
10.5.7 Dislocation Kinetics in the High-Temperature Range . . . 453
10.5.8 TheClimb-ExchangeModel ........................455
10.6 ConclusionsonQuasicrystals..............................459
11 Conclusion ................................................463
List of Abbreviations and Symbols .............................465
List of Video Clips .............................................471
References .....................................................475
Index ..........................................................499
Part I
General Properties of Dislocation Motion
1
Introduction
The properties of crystalline solids can be classified into two groups. To the
first one belong, for example, the elastic properties or the occurrence of X-ray
diffraction patterns. These phenomena are controlled by the regular periodic
structure of the crystal lattice. The second group involves properties like diffu-
sion or the mechanical strength. Though these phenomena are also influenced
by the regular crystal structure and its binding properties, they are essen-
tially controlled by the defects in the regular arrangement of the atoms and
may be called structure sensitive properties, as was introduced by Smekal [1].
The crystal defects are classified according to their extension in space into
zero, one, two, and three-dimensional defects. Zero-dimensional defects or
point defects include single missing atoms called vacancies, interstitial atoms,
and substitutionally or interstitially incorporated foreign atoms like impuri-
ties. One-dimensional defects occur if the regular coordination of atoms is
disturbed along a line. These defects are called dislocations being the topic
of this book. Two-dimensional defects are grain and phase boundaries while
three-dimensional ones are larger precipitates or inclusions. Since the move-
ment of dislocations is influenced by all types of defects, it turns out to be a
very complex process.
In the following section, it is shown that shearing a crystalline solid along a
plane as a whole requires a high stress, which exceeds the measured strengths
of materials by several orders of magnitude. Afterwards, in Sect. 1.2, dislo-
cations are introduced. Their motion through the crystal lattice allows the
shearing in small steps, yielding realistic values of the mechanical strength.
Part I of the book treats the general features of the dynamic dislocation
behavior. At first, an outline of the experimental methods is given in Chap. 2.
As a basis for understanding the dynamics, the geometric and elastic dislo-
cation properties as well as the structure of dislocations in real crystals are
reviewed in Chap. 3 before the dislocation motion itself is treated in Chap. 4.
Chapter 5 is concerned with the kinetic processes of dislocation generation,
immobilization and annihilation. Part II is devoted to the dislocation motion
in particular material classes, that is, to semiconductors in Chap. 6, ceramic
4 1 Introduction
single crystals (Chap. 7), metals (Chap. 8), intermetallic alloys (Chap. 9),
and finally to quasicrystals (Chap. 10). In these chapters, particular disloca-
tion processes are discussed in more detail and, whenever possible, they are
illustrated by video sequences.
1.1 Theoretical Yield Strength
In the 1920s, several authors tried to estimate the yield strength, that is, the
critical stress necessary to plastically extend or shear a crystalline body. The
following estimation of the shear strength is based on a work by Frenkel [2].
As sketched in Fig. 1.1, a shear stress τ acts on the solid and tries to shift the
lattice planes A and B of distance h with respect to each other. The distance
between the atoms within the planes is b. This is the periodicity distance of
the lattice. Originally, the planes are in elastic equilibrium. The stress acting
tries to shift the upper plane B in the x direction. After a shift by b, the body
is in equilibrium again. It is assumed that the stress necessary to shift plane
B is a sine function of x
τ = K sin(2πx/b).
K is an interaction strength between the planes, which has to be deter-
mined in agreement with a macroscopic quantity. In this case, it is the shear
modulus μ. It is defined as
μ =
dτ
dγ
x=0
,
where γ is the (elastic) shear strain. In Fig. 1.1, the increment in γ is given
by dγ =dx/h.Thus,
μ = h
dτ
dx
x=0
=
2πhK
b
cos(2πx/b).
For x =0,K becomes
K =
μb
2πh
,
h
h
A
B
dx
d
γ
b
x
τ
τ
Fig. 1.1. Model for calculating the theoretical shear strength
1.2 Plastic Shear by the Motion of Dislocations 5
Table 1.1. Shear modulus and real (normal) yield strength
Material μ (MPa) σ
y
(MPa)
Al alloys 27,000 100 . . . 500
Steels 86,000 400 ...1,500
or
τ =
μb
2πh
sin(2πx/b).
The maximum value is obtained for x = b/4
τ
id
=
μb
2πh
.
This is an estimate of the shear strength of an ideal crystal lattice if it is
sheared along a plane as a whole. It may be noted that this strength depends
on the ratio between the periodicity distance b and the spacing h between
the sheared planes. With b/h =1/
√
2 in the f.c.c. lattice, it follows that
τ
id
= μ/(π
√
8) ≈ μ/10. Some values of the shear modulus and the measured
technical yield stresses σ
y
of materials are given in Table 1.1. The relation
between the shear stress and the technical (normal) stress will be described
in Sect. 2.1.
As a conclusion, the real yield stresses of materials are about two orders of
magnitude smaller than the theoretical strength of an ideal crystal lattice. The
difference between both is due to the existence of crystal defects, primarily of
dislocations. The dynamic behavior of the latter determines the characteristics
of plastic deformation.
1.2 Plastic Shear by the Motion of Dislocations
The great difference between the experimental yield stresses and the theoret-
ical shear strength necessary to shift two lattice planes with respect to each
other as a whole can be explained by the occurrence of line defects called
dislocations in a crystalline solid and their motion under stress. The concept
of crystal dislocations was introduced by Orowan [3], Polanyi [4], Taylor [5],
and Burgers [6]. The elastic stress and strain fields of dislocations in an elastic
continuum were calculated before by Volterra [7].
The process of the motion of a dislocation through a crystal is illustrated
in Fig. 1.2. The undisturbed crystal is shown in Fig. 1.2a. The vertical lines
inside the body mark the lattice planes. The acting shear stress τ has created
a step on the left surface by breaking bonds in the left outer plane and by
pushing the upper half-plane into the crystal (Fig. 1.2b). The lattice is intact
everywhere except along the line D at the inner end of the inserted extra half-
plane. This line is the place of the dislocation. In addition to the disturbance
6 1 Introduction
τ
(a)
(b)
(c)
(d)
b
τ
D
D
Fig. 1.2. Shearing by the motion of a dislocation through a crystal
of the coordination of the atoms near the dislocation line, the whole lattice is
elastically strained, characterized by bent lattice planes. Under the action of
the stress, the dislocation may move through the crystal as in Fig. 1.2c and
finally emerge on the right outer plane, producing a surface step there also as
in Fig. 1.2d. After the whole process, the body is sheared along the plane on
which the dislocation moved by the vector b, that is, it permanently changed
its shape owing to a plastic strain. The shift vector b is called the Burgers
vector and describes not only the shift of one part of the body with respect
to the other, but also the intensity of the strain field around the dislocation.
In the case of Fig. 1.2, b is perpendicular to the dislocation line.
Figure 1.3 presents a high-resolution electron micrograph of a dislocation
in a real crystal, a thin PbTe foil with a (001) foil plane. The bright dots
represent columns of atoms. The PbTe foil has the NaCl structure, that is,
the distance between the cube planes is half the lattice constant. The dislo-
cation in the center becomes visible when the image is viewed at a glancing
angle. Viewing from the upper left corner in diagonal direction reveals that,
according to the crystal structure, two extra half-planes are inserted from
below and end at the location of the dislocation. The latter extends perpen-
dicular to the image plane. As shown in the image, a dislocation involves the
disturbance of the regular arrangement of atoms along a line, which causes
a far-reaching elastic strain field, visible as a bending of the atom planes. In
the diffraction contrast mode of transmission electron microscopy, the strong
strain field near the dislocation lines is imaged and the dislocations appear
as lines as in Fig. 1.4. The figure shows dislocations that have formed during
plastic deformation and which are arranged on different planes.
A very important quantity for characterizing a dislocation microstructure
is the dislocation density . It is defined as the length of dislocation lines
per volume. Frequently, the dislocation density is measured as the number of
emergence points through a surface. Both quantities are roughly equal except
1.2 Plastic Shear by the Motion of Dislocations 7
2.5 nm
Fig. 1.3. High-resolution electron micrograph of a dislocation in a thin PbTe foil.
The image plane is a (001) plane. Micrograph by Roland Scholz
for a numerical factor close to unity, which depends on the geometry of the
dislocation arrangement. Characteristic densities are given in Table 1.2.
It may be noted that the dislocation densities in strongly deformed mate-
rials correspond to 1–1,000 km mm
−3
. On the other hand, the concentration
of atoms with a disturbed neighborhood is quite low, as the following estima-
tion shows. The number of atoms with disturbed coordination is about 5 per
lattice plane, that is, 5/a per unit length of the dislocation or 5/a per unit
volume. Here, a is the lattice constant. Considering that the total number
of atoms per unit volume is 1/a
3
, the concentration of atoms of disturbed
coordination is about 5a
2
. With characteristic values of =10
12
m
−2
and
a =3×10
−10
m, the concentration becomes about 5 ×10
−7
. This concentra-
tion is not very high, and it is in the order of magnitude of that of intrinsic
thermal point defects at about 70% of the absolute melting temperature.
The plastic deformation realized by the motion of dislocations is gov-
erned by their geometric properties, which will be described in Sect. 3.1. The
dynamic properties are controlled by the interactions with the crystal lattice
itself as well as with all the other defects in the material. The latter inter-
actions are essentially of elastic nature. Therefore, the elastic properties of
dislocations are treated in Sect. 3.2. Finally, the features of the dislocations
are also influenced by the crystal lattice structure. This will be described in
8 1 Introduction
1 µm
g
Fig. 1.4. Dislocation structure in a cross-sectional foil of an FeAl single crystal
deformed about 2.5% at 533
◦
C and 564
◦
C. g =(1
¯
10) is the imaging diffraction
vector near the pole n = [111]. Inset: Equilibrium shape of dislocations with [010].
Burgers vector on a (101) plane projected onto the image plane, calculated by the
line tension model using anisotropic elasticity theory as described in Sect. 3.2.7.
From [8]. Copyright Elsevier Ltd. (2005)
Table 1.2. Characteristic dislocation densities in some materials
Material (m
−2
)
Semiconductor single crystals 0 or very few
Well annealed metal crystals 10
7
...10
8
Slightly deformed crystals 10
10
...10
12
Strongly deformed crystals 10
12
...10
15
Sect. 3.3. These topics are the basis of studying the dislocation motion and
dynamics in Chap. 4.
For a detailed study of particular aspects of dislocation dynamics, the
following textbooks are recommended. The thermodynamics of dislocation
motion are treated in a fundamental way in the book Thermodynamics and
Kinetics of Slip by Kocks, Argon and Ashby (1975) [9]. Several mechanisms
of dislocation dynamics and many experimental data are given in Dislocation
Dynamics and Mechanical Properties of Crystals by Nadgornyi (1988) [10].