Messerschmidt U. Dislocation Dynamics During Plastic Deformation

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1.2 Plastic Shear by the Motion of Dislocations 9

The thermally activated dislocation motion is described in Thermally Acti-

vated Mechanisms in Crystal Plasticity by Caillard and Martin (2003) [11].

Finally, many aspects of the theory of dislocations are presented in detail in

the textbook Theory of Dislocations by Hirth and Lothe (1982) [12], which is

quoted frequently in the present book. Before the dislocation properties will

be discussed in detail, the next chapter describes the experimental methods

that mainly yield information on the mechanisms of dislocation dynamics.

Experimental Methods

Information on the dynamic dislocation behavior can be obtained by methods

on diﬀerent scales. This ranges from the measurement of macroscopic defor-

mation data to microscopic techniques like transmission electron microscopy

and, in particular, to in situ deformation tests in the electron microscope.

Several methods have frequently been used and are described below in some

detail. Others are only of supplementary character.

2.1 Macroscopic Deformation Tests

This section introduces the measurement of the macroscopic ﬂow stress and

its sensitivity to changes in the temperature and the deformation rate. These

quantities will be used in Sect. 4.1 to determine the thermodynamic parame-

ters of the dislocation motion. If a body of homogeneous cross section A and

length l is loaded by a force F , it undergoes a change in length Δl.Itisuseful

to normalize the force by the cross section and the length change by the length

to obtain the (technical) tensile or compression stress σ and total strain ε

σ = F/A, ε

=Δl/l. (2.1)

The total strain may be composed of the reversible elastic strain ε

and the

irreversible plastic strain ε

= ε

+ ε. (2.2)

The elastic strain is determined by the laws of elasticity. Under most experi-

mental conditions, a plastic strain in a crystalline solid is realized by shearing

the material in crystallographic directions along crystallographic planes as

outlined in Fig. 2.1.

The plastic shear deformation is caused by a shear stress τ resolved onto

the shear plane, which results from the applied stress σ according to Schmid’s

law [13]

τ = m

σ =cosϕ cos ψσ, (2.3)

12 2 Experimental Methods

Fig. 2.1. Plastic shearing of a crystalline solid

where ϕ is the angle between the direction of shear b andthespecimenaxis,

and ψ is that between the normal of the shear planes n and the specimen axis.

The product of both cosines m

is called the orientation factor (in German

Schmid-Faktor). The direction of the shearing and the shear plane deﬁne a so-

called slip system. Similarly, there is a relation between the (normal) plastic

strain ε and the plastic shear strain γ =tanδ (see Fig. 2.1)

γ = ε/m

. (2.4)

During the deformation, the specimen cross section and the orientation of

the shearing direction and of the shear planes change. For large deformations,

this should be taken into account in (2.3) and (2.4). In this book, mostly

small deformations are discussed. It is therefore suﬃcient to consider only the

starting data.

The relevant outer parameters controlling the plastic deformation are the

temperature T , the plastic strain rate ˙ε, and the stress σ. In most experiments,

the temperature is kept constant, at least for an interval of time. In addition,

one of the other two parameters remains constant, too. If this is the stress

(or in many cases simply the load F ), the deformation rate will change in

time, and the experiment is then called a creep test. In the other case, the

deformation machine enforces a certain strain rate and the stress becomes a

function of strain or time. Mostly, the total strain rate is controlled by the

deformation machine so that the elastic strains in (2.2) include also the elastic

deformation of the machine or, at least, parts of it. Thus, the plastic strain

rate may diﬀer more or less from the nominal one, as described later. The

experiments with a constant strain rate are called quasistatic experiments.

Most experiments described in this book are of this type. Special machines

are designed to reach very high strain rates, for example, by using impact

loading. In these dynamic experiments, neither the stress nor the strain rate

and even not the temperature remain constant.

The typical result of a quasistatic deformation test is the stress–strain

curve. Two examples are outlined in Fig. 2.2, where the actual applied stress

σ necessary to deform the specimen is plotted vs. the strain, in the simple

2.1 Macroscopic Deformation Tests 13

Fig. 2.2. Stress–strain curves of single crystals. (a) Nb single crystal deformed at

room temperature with strain rate cycling tests SRC between 10

−6

and 10

−5

−1

(SRC1toSRC4)andupto10

−4

−1

(SRC5). (b)MoSi

single crystal deformed at

two temperatures along a 110 orientation at a strain rate of 10

−5

−1

.(c)Section

of a stress–strain curve of an Al-Pd-Mn single quasicrystal deformed at 755

◦

Cwith

a strain rate change (SRC) from 10

−4

to 10

−3

−1

. Data from [14–16]

14 2 Experimental Methods

case the total strain ε

. The curve in Fig. 2.2a starts with the elastic range

ER. Its slope should describe the elastic modulus (stiﬀness modulus) S of

the specimen plus those parts of the deformation machine which are located

within the strain measuring device. In practice, the connecting elements

between the specimen and the machine are not ideal. For example, the grip

faces of the specimen are neither fully plane nor parallel in a compression

test, so that the specimen and the machine are contacted only in a small

part of the specimen cross section. This part experiences a high load caus-

ing plastic deformation before the whole specimen deforms. This results in

a reduced slope of the “elastic” line. It is therefore useful to determine the

stiﬀness modulus S from the unloading curve UL at the end of the experiment

where irreversible plastic deformations are strongly reduced.

At a certain stress, the specimen starts to deform plastically, leading to

a decrease of the slope of the curve. Under the actual deformation condi-

tions, the plastic strain rate depends on the stress and the “microstructure”

of the specimen. As described earlier, the dislocation density is an important

quantity characterizing the microstructure. The stress necessary to continue

the deformation is the ﬂow stress σ. The threshold stress for plastic defor-

mation is called the yield stress σ

.IncurvesasthatinFig.2.2a,σ

can

be determined by linear extrapolation of the plastic range onto the elastic

line. Frequently, the stress–strain curve goes through a maximum followed by

a minimum called the upper and lower yield points (UYP and LYP) as in

Fig. 2.2b. This behavior results from drastic changes of the microstructure,

mainly a strong increase in the dislocation density. In this case, the UYP can

be taken for σ

. It is an essential aim of this book to interpret the yield stress

on the basis of the dynamic dislocation behavior. After the yield point, one

or several ranges of plastic deformation follow with an increase in the ﬂow

stress. This increase is called work-hardening. The slope of the ﬂow stress vs.

the plastic strain Θ =dσ/dε is called the work-hardening coeﬃcient.

With S =dσ/dε

, it follows from (2.2) that

˙ε

=˙ε

+˙ε =˙σ/S +˙ε. (2.5)

Thus, the plastic strain rate is diﬀerent from the total or nominal strain rate

of the deformation machine, namely

˙ε =˙ε



1 −

dσ

dε



=˙ε

S + Θ

. (2.6)

Both are equal only at constant stress or Θ = 0, that is, at the UYP and

LYP and in ranges of low work-hardening, that means during steady state

deformation.

To determine thermodynamic quantities characterizing the dislocation

dynamics, the sensitivity of the ﬂow stress with respect to changes in the

plastic strain rate r =Δσ/Δln ˙ε and the temperature Δσ/ΔT have to be

studied. An early review of this topic is given in [17]. On principle, this can

2.1 Macroscopic Deformation Tests 15

be done by measuring stress–strain curves at diﬀerent strain rates and temper-

atures and forming the respective diﬀerence quotients of the strain rate and

temperature dependencies of the yield stress. However, this has the disadvan-

tage that specimens deformed under diﬀerent conditions may exhibit diﬀerent

microstructures so that the diﬀerences are not only due to the change in the

outer variables. It should therefore be preferred to perform instantaneous

changes of the deformation parameters during the deformation experiments

and to observe the transient behavior. It is assumed that changes in the

microstructure need strain or time so that the microstructure remains essen-

tially constant during the sudden changes of the deformation conditions and

that the transients can be observed. Experiments with instantaneous changes

of the strain rate are called strain rate cycling tests (SRC). Such tests with

increases and decreases of the strain rate by factors of 10 are included in the

stress–strain curve of Fig. 2.2a. After the changes, the microstructure adjusts

to a new state, leading to a yield drop eﬀect similar to that at the begin-

ning of the deformation. It is now possible to deﬁne two stress increments

shown in Fig. 2.2c, one using the maximum stress value Δσ

corresponding

roughly to the instantaneous change in the strain rate and one describing

the diﬀerences in the steady state values of the ﬂow stress Δσ

. In the SRC

tests in Fig. 2.2a, the transient eﬀect is very weak but is obvious at SRC6.

The situation is diﬀerent when the temperature has to be changed (TC). In

most deformation machines, some time (in the order of 30 min) is necessary

to adjust the machine to new steady state conditions. Thus, instantaneous

temperature changes are not possible. Besides, the specimen and parts of the

machine change their length so that the specimen has partly to be unloaded

before the temperature can be changed. Accordingly, the stress increments

Δσ

usually do not refer to instantaneous changes.

In addition to strain rate cycling experiments, the strain rate sensitivity

r can well be determined by so-called stress relaxation tests, indicated by R

in Fig. 2.2b. During the deformation, the testing machine is instantaneously

stopped, that is, ˙ε

= 0 in (2.5). Since the stress is still acting, the specimen

continues to deform at decreasing stress according to

˙ε = −˙σ/S. (2.7)

Thus, the rate of the relaxing stress is proportional to the strain rate. The

strain rate sensitivity is then given by

r =

∂σ

∂ ln ˙ε

∂σ

∂ ln(−˙σ)

. (2.8)

For these experiments, it is necessary only to measure the time dependence

of the relaxing stress and to plot the logarithm of the relaxation rate vs.

the stress, as demonstrated by curve R in Fig. 2.3. Except for a shift along

the ordinate axis, these curves describe the dependence of the strain rate

on the stress. To scale this relation, the stiﬀness modulus S is required, which

16 2 Experimental Methods

Fig. 2.3. Stress relaxation curve R and repeated stress relaxation curve RR of an

Al-Pd-Mn single quasicrystal deformed at 818

◦

C. Data from [16]

can be taken from the unloading curve, as described earlier. The inverse slope

of the relaxation curves equals the strain rate sensitivity r. If the slope is deter-

mined at the beginning of the curve, the strain rate sensitivity corresponds to

the strain rate just before starting the relaxation test. The dependence of r on

the stress is obtained by measuring slopes at diﬀerent stresses along the curve.

Stress relaxation tests consume far less strain than strain rate cycling tests so

that microstructure changes should be less severe. The strain and time neces-

sary to relax a certain stress interval depend on the stiﬀness modulus S.Itis

therefore desirable to use a machine of high stiﬀness. This can be achieved by

electronic strain control with the strain gauges attached near the grip faces of

the specimen as close as possible. Nevertheless, the microstructure can change

during relaxation tests. This can be studied by comparing the initial relax-

ation curve (R in Figs. 2.2 and 2.3) with a repeated one RR obtained after

stopping the machine during reloading before the original stress and strain

rate are reached again.

Alternatively, experiments with instantaneous changes of the deformation

conditions can also be performed during creep tests. The counterpart to strain

rate change tests are then load change tests where the strain rate goes through

a transient and reaches a new value.

2.2 Stress Pulse Double Etching Technique

One of the most important topics of dislocation dynamics is the determination

of the stress dependence of the dislocation velocity. Many data were collected

in the 1960s and 1970s using the stress pulse double etching method. This

technique was introduced by Gilman and Johnston [18,19] and consists of the

following steps:

• Decreasing the dislocation density of the starting single crystals by long-

time annealing at a high temperature

2.2 Stress Pulse Double Etching Technique 17

• Cutting suitable specimens

• Removing the surface damage from cutting by careful chemical or electro-

chemical polishing

• Introduction of fresh dislocations

• Marking the starting positions of the dislocations by chemical or electro-

chemical etching

• Moving the dislocations by applying a stress pulse

• Second etching to mark the ﬁnal positions of the dislocations after the

stress pulse, and calculation of the velocity from the displacement and the

duration of the stress pulse.

The fresh mobile dislocations have been introduced by diﬀerent methods.

Johnston and Gilman used a steel ball pressed slightly onto the crystal surface.

In their ﬁrst study of the dislocation mobility in a metal, Stein and Low [20]

scratched the surface with a steel knife. Later on, also micro-hardness indents

were used to produce fresh dislocations (e.g., [21]). When many dislocations

are introduced by scratching or indenting, the dislocations should be moved

away from the region of high internal stresses connected with the concen-

trated load by a ﬁrst stress pulse before marking their starting positions. This

requirement was not taken into account in several measurements.

In many cases, the pyramidal shape of the etch pits reﬂects the symme-

try of the etched crystal face, as shown in Fig. 2.4 for the 100 face of the

LiF crystals used in the original study of Johnston and Gilman. When the

dislocation leaves the starting etch pit by the stress pulse, the etch pit does

not grow in depth during further etching so that the original pit gets ﬂat at

its bottom as demonstrated in the left pit and the central pit in the ﬁgure,

where the dislocation moved in two steps. Instead, a new smaller pit develops

in the ﬁnal position of the dislocation after the stress pulses (the right pit in

the ﬁgure). The etch pits are mostly imaged by usual optical or interference

microscopy or by scanning electron microscopy.

10 µm

Fig. 2.4. Selective etching of a dislocation moved in two steps from left to right.

From Gilman and Johnston [18]. Copyright Wiley–Blackwell (1957)

18 2 Experimental Methods

The stress pulses are usually applied in the bending geometry either in

4-point (e.g., [19]) or 3-point bending (e.g., [20]). Four-point bending has the

advantage of an equal stress distribution within the inner span of the bending

jig and of the opportunity to use both the tension and compression faces

to study a tension–compression asymmetry. In 3-point bending, the bending

moment decreases from the center outwards so that, with several scratches or

indents, the stress dependence of the dislocation velocity can be studied over a

certain range on a single specimen. As the plastic strains are small during the

experiment, the shear stresses are calculated from the bending load applying

common elasticity theory.

The duration of the stress pulses is chosen such that the displacements

range between a few micrometers and about one hundred micrometers, which

can easily be observed by the etching techniques. This requires, depending on

the stress, durations between days and 10

−6

s to reach dislocation velocities

between 10

−11

and 10

−1

. These pulses are produced by materials testing

machines down to seconds or fractions of a second, by magnetic systems similar

to loudspeaker drives down to the millisecond range, and by impact loading

for shorter times.

In the best case, the starting and ﬁnal positions of a dislocation can indi-

vidually be coordinated as in Fig. 2.4. When the dislocations move out of a

scratch or an indent, this is frequently not possible. Individual dislocations

may multiply on their way, or a source may emit many dislocations moving on

the same or neighboring planes. As a consequence, the leading (head) disloca-

tion moves under the action of not only the applied stress but also the internal

stress ﬁeld of the other dislocations moving behind. Thus, its velocity will be

higher than that of an isolated single dislocation. This eﬀect is discussed in

Sect. 5.2.2, also showing that the behavior of groups of dislocations cannot

unambiguously be extended to that of individual dislocations.

In spite of some limitations of the etching method, a considerable bulk of

data has been measured by this technique. Almost all data available at that

time have been reviewed by Nadgornyi [10].

2.3 Transmission Electron Microscopy

Transmission electron microscopy (TEM) is the most powerful method for

studying the mechanisms of dislocation motion. In TEM, diﬀerent modes of

imaging are possible, depending on the number of diﬀracted beams contribut-

ing to the formation of the image. Each diﬀracted beam originates from the

reﬂection of electrons at a set of lattice planes. In high-resolution electron

microscopy, many reﬂected beams contribute to the image formation. Accord-

ingly, individual atom columns can be resolved if the resolving power of the

electron microscope is suﬃcient. A respective micrograph of a dislocation was

shown in Fig. 1.3. Many processes controlling dislocation dynamics do not

only occur on the atomic level but also on a scale between some nanometers

2.3 Transmission Electron Microscopy 19

PPR R'

Fig. 2.5. Formation of diﬀraction contrast at a dislocation

and several micrometers. A suitable mode of imaging is then the so-called

diﬀractioncontrast.Inthismode,thespecimenisorientedinsuchawaythat

the diﬀracted intensity originates from a single set of lattice planes, as out-

lined in Fig. 2.5. A great part of the intensity of the incident electron beam

passes the specimen as the primary beam P. The other part is reﬂected at the

respective lattice planes forming the reﬂected beam R as shown for the left

beam. If these planes are bent owing to the presence of a dislocation as for the

right beam, the reﬂected beam points into another direction R



, resulting in

an electron microscopy contrast, if the imaging conditions are adjusted prop-

erly. Thus, the contrast results from the elastic strain ﬁeld around the core

of the dislocation. A respective micrograph in bright-ﬁeld contrast with the

dislocations imaged as dark lines on a bright background was already shown

in Fig. 1.4.

It is not possible to present the details of diﬀraction contrast image for-

mation within the framework of this book. The reader is referred to standard

works of TEM in materials science like [22–26]. Only the method of contrast

extinction for determining the direction of the Burgers vector introduced in

Sect. 1.2 is brieﬂy mentioned here. As may be inferred from Fig. 2.5, the planes

perpendicular to the Burgers vector b are most strongly distorted due to the

presence of the dislocation. The planes containing the Burgers vector like the

image plane of the ﬁgure are not or only weakly bent. If the image is formed by

exciting these planes, the contrast of the dislocation is extinguished. The set

of imaging planes is characterized by its diﬀraction spot and by its reciprocal

lattice vector g. The contrast extinction therefore requires that the vectors g

and b are perpendicular to each other, or that their scalar product is zero

g · b =0. (2.9)

To obtain the direction of the Burgers vector in space, it is necessary to

ﬁnd two extinction conditions with independent g vectors for the respective