Messerschmidt U. Dislocation Dynamics During Plastic Deformation

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160 5 Dislocation Kinetics, Work-Hardening, and Recovery

in opposite directions. The parameters were chosen to ﬁt the experimen-

tally observed diﬀerences between the velocity of individual dislocations and

head dislocations in slip bands in NaCl single crystals [273]. According to

this work, the head dislocations do not reach a constant velocity. Instead,

the velocity decreases with increasing time. This underlines the problem of

deducing velocity–stress relations from measurements on head dislocations in

dislocation groups.

The dynamic behavior of localized dislocation sources during plastic

deformation in regions with a high dislocation density or in polycrystalline

materials with slip distances limited by grain boundaries diﬀers from that

just described. Related experimental observations will be mentioned in the

next section.

The second mechanism of dislocation generation, which creates individual

isolated dislocation loops, is the double-cross slip mechanism suggested by

Koehler [274] and Orowan [275] and experimentally ﬁrst observed indirectly

by Johnston and Gilman [19, 276]. It is shown schematically in Fig. 5.5. A

screw dislocation (a) moves on its glide plane, which is identical with the

image plane. Cross slip of a segment of length L results in two superjogs J

(b). They can move conservatively along the dislocation line but not in the

forward direction of the dislocation so that they act as pinning agents, and the

segments near the jogs bow out. The segment between the jogs then multiplies

similarly to the Frank–Read source (c).

The sources can act only if certain criteria are fulﬁlled. The length of the

bowing segment has to be larger than the critical length L

to overcome the

half-circle conﬁguration of (3.47). Inserting characteristic values of μ = 30–

100 GPa, b =3× 10

−10

m, and τ = 100–1000MPa results in L

to be in the

order of magnitude of 100 nm. In stage (c) of the double-cross slip mechanism,

the branches adjoining the jogs have to pass each other on their parallel planes.

This is only possible if the height of the jogs, that is, the distance between

the parallel glide planes, is larger than a critical value h

given by the dipole

opening criterion (3.27). It is obvious that h

is about 20 times smaller than

. Thus, both criteria involve a dependence of the dislocation generation

processes on the stress.

(a) (c) (d)(b)

Fig. 5.5. Double-cross slip mechanism of the generation of individual dislocation

loops and of the production of dislocation debris

5.1 Dislocation Kinetics 161

If the jogs are not high enough to fulﬁl the dipole opening criterion, dis-

location dipoles are trailed by the moving dislocation as outlined in Fig. 5.5d.

The dipoles can be terminated by conservative motion of the jogs away from

the dipoles. The dislocation dipoles formed during the motion of screw disloca-

tions are also called dislocation debris. The accumulation of the debris during

plastic deformation hardens the crystals. After the ﬂow stress has increased,

originally stable dipoles can fulﬁl the dipole opening criterion and open to

form new dislocation loops.

Experimental values of the frequency of multiplication events can be

concluded from the statistical data on cross slip processes in NaCl single

crystals. The frequency distribution H(h) of the cross slip heights h was given

in Sect. 4.3 (Fig. 4.10). Only those cross slip events with h>h

are rele-

vant for multiplication. The latter fraction of the total cross slip density D

(Fig. 4.11) determines the density of cross slip events with suﬃcient height for

a multiplication

= DΔh

∞



H(h) .

Δh =0.053 μm is the width of the columns in Fig. 4.10. Using (3.27) and the

corresponding data of NaCl single crystals, μ =18GPa,b =3.96 × 10

−10

the ﬂow stress τ =1MPa,andν =0.3, the critical cross slip height for mul-

tiplication becomes h

=0.4 μm. The total density of cross slip events along

the path of the moving dislocation is D =38mm

−1

. The tail of the histogram

with h>h

(Δh times the sum) corresponds to a fraction of 0.022. Thus,

=0.8mm

−1

, or a dislocation has to travel an average distance of about

1.2 mm before a multiplication occurs. This large distance is in accordance

with the very long slip lines of individual dislocations on the NaCl single crys-

tals. As all parameters involved, that is, the frequency distribution of cross

slip heights H(h), the critical height for multiplication h

, and the total den-

sity of cross slip events D, depend on the stress, the multiplication rate may

be assumed to be proportional to the stress as supposed below.

and h

are well below the foil thickness of about 500 nm in an HVEM

in situ experiment, so that the mechanisms of dislocation generation can well

be observed. The intermediate conﬁguration of stage (c) is often metastable.

It is characterized by the highlighted α-like conﬁguration in Fig. 5.5c, which

is frequently observed in dislocation structures under stress. The double-cross

slip mechanism usually emits only a single new dislocation loop. As the gen-

erated dislocation moves on a plane parallel to the original one, slip may

spread leading to the width of the slip bands to grow, which is in contrast

to the Frank–Read source, which emits many dislocations on the same or a

neighboring plane.

Cross slip events occur during the motion of dislocations. Accordingly, the

increase in the dislocation density d

should be proportional to the area dA

swept by all dislocations as suggested in [277]. Considering the dependence of

the creation rate of dislocations on the stress, discussed above, the creation

162 5 Dislocation Kinetics, Work-Hardening, and Recovery

rate may be written as

d

= wτ

∗

dA = wτ

∗

ds =(w/b)τ

∗

dγ, (5.2)

where the multiplication coeﬃcient w is a constant, ds is the displacement of

all dislocations, and dγ is the increment in shear strain. The multiplication is

driven by the local eﬀective stress τ

∗

rather than by the total stress τ. Such

an expression was used by Haasen and Alexander [278, 279] to explain both

the initial increase of the dislocation density in semiconductor single crystals

and the related yield drop eﬀect.

5.1.2 Experimental Evidence of Dislocation Generation

As mentioned earlier, dislocation generation can well be studied during in situ

straining experiments in the TEM. An overview is given in [280]. In several

materials, new dislocations are generated in localized (Frank–Read) sources.

Mostly only half the operating segment of the sources in Figs. 5.2 and 5.5

is visible, as in Fig. 5.6 taken in duplex steel. The other half is cut away by

the specimen surface. Then, the operating segment rotates around the single

pinning agent, forming a new dislocation loop during each revolution. The

ﬁgure presents three stages where always one dislocation is emitted between

the stable stages. A video sequence of the same source is given in Video 8.8.

Video 9.16 shows a similar source in an MoSi

single crystal.

In contrast to the dynamic behavior of the Frank–Read source described

in the preceding section, which emits dislocations into the undisturbed crys-

tal volume, the dislocations emitted from the present sources pile up either

against grain or phase boundaries or against groups of other dislocations. As

will be discussed in Sect. 5.2, the pile-ups produce a back stress on the source,

which opposes the applied stress. As a consequence, long-range the source

stops operating and starts again only after one of the outer dislocations is

released from the pile-up, thus reducing the back stress.

Fig. 5.6. Three stages of the operation of a localized dislocation source (Frank–

Read source) in an austenite grain during the in situ deformation of duplex steel

taken from a video recording. Frames from [280]

5.1 Dislocation Kinetics 163

In many materials, the new dislocations are generated by the double-cross

slip mechanism. One example showing all the aspects of this mechanism is

the deformation of MgO single crystals at room temperature [281]. Figure 5.7

presents several stages of the opening of a new dislocation loop at a jog in a

screw dislocation. If the jog is not high enough for the loop to open (h<h

a dislocation dipole is trailed behind the jog. Figure 5.8a shows a dislocation

with a jog, which is indicated by a small trailed dipole (arrow). In Fig. 5.8b, the

dislocation has proceeded further making the trailed dipole well visible. The

dipole and the upper part of the dislocation are moving during the exposure

of the ﬁlm. In Fig. 5.8c, the dipole has achieved a stable position now marked

by strong contrast. Finally, the dipole has terminated in d, that is, the jog

again had glided along the dislocation, thus separating the dipole from the

dislocation. In some materials, many dipoles are created during plastic defor-

mation. These processes were ﬁrst observed by Johnston and Gilman [276]

and may be responsible for part of work-hardening.

While the deformation is going on, the stress may increase because of

work-hardening. As a consequence, dipoles having formed during earlier stages

of the deformation may now open as the bypassing criterion (3.27) is now

1 µm

Fig. 5.7. Formation of a new dislocation by the opening of a loop at a jog during in

situ deformation of an MgO single crystal at room temperature. b is the projection

of the Burgers vector. Micrographs from [281]

1 µm

(d)

(c)

(a)

(b)

Fig. 5.8. Trailing of a dislocation dipole at a jog in a moving screw dislocation in

MgO. Micrographs from [281]

164 5 Dislocation Kinetics, Work-Hardening, and Recovery

(a) (b)

0.5 μm

Fig. 5.9. Opening of dipoles to form glide loops during in situ deformation of an

MgO single crystal. Micrographs from [281]

fulﬁlled. This is shown by two examples in the sequence of Fig. 5.9. In Fig. 5.9b,

dipole A opens to a loop. The loop of dipole B emerges partly through the

surface so that a half-loop remains. In Fig. 5.9c, loop A opens further and

emerges through both surfaces in d leaving a pair of new dislocations A1 and

A2. Several dislocation multiplication processes are demonstrated in Video 7.1.

A survey of the data on dislocation generation from the in situ defor-

mation studies performed in Halle is presented in Table 5.1. Although these

experiments reveal the diﬀerent mechanisms of dislocation generation, the

understanding on a quantitative level is still poor. In contrast to the localized

Frank–Read sources, dislocation multiplication requires cross slip. While in

the f.c.c. metals the cross slip planes are of the same type as the primary

slip planes resulting in an equal dislocation mobility, in most other materials

the mobility on the cross slip planes is lower than on the primary ones. In

ZrO

-10 mol% Y

, for example, the ﬂow stress on the cross slip planes is

about 30% higher than that on the primary ones [289]. The ﬂow stress ratio

between both planes certainly inﬂuences the width of the cross slip events.

As described earlier, the double-cross slip mechanism leads to the growth of

slip bands by the sidewise spreading of slip. Since double-cross slip events of

smaller height may lead to multiplication at higher stresses, the slip bands

become narrower in strong materials and at low temperatures.

In NaCl single crystals, cross slip is initiated by long-range internal stress

ﬁelds. Only in materials where extended dislocations have to constrict before

cross slip, the initiation process is thermally activated. The multiplication

5.1 Dislocation Kinetics 165

Table 5.1. Burgers vectors, slip and cross slip planes as well as type of dislocation

generation in diﬀerent materials studied at the Halle HVEM facility and summarized

in [280]

Material, load

axis

Burgers vector Slip planes Cross slip

planes

FR DCS

Si, Ge 1/2110{111}{111} [163]

NaCl, 100 1/2110{110}{100} [179]

MgO, 100 1/2110{110}{122}, {211} [281]

ZrO

, 112 1/2110{100}{110}, {111} [282]

Al–Zn–Mg 1/2110{111}{111} [226]

Al–Ag 1/2110{111}{111} [283]

Al–8Li 1/2110{111}{111} [284]

Duplex steel

austenite

1/2110{111} [280]

Duplex steel

ferrite

1/2111{110}, {112},

{123}

{110}, {112},

{123}

[280]

Ti–6Al a/311

20{0001},

100}

{0001},

100}

[285] [285]

γ–Ti–52Al 1/2110{111}{111} [286]

NiAl, 100100{100}, {110},

{210}

{100}, {110},

{210}

[287]

MoSi

, 210 1/2111{110} [288]

FR localized Frank–Read source, DCS double-cross slip mechanism

event, that is, the bowing after (3.47) and the bypassing after (3.27) are also

athermal processes. This is conﬁrmed by the very instantaneous character of

multiplication in ZrO

where the ﬂow stress is alsoofathermalnatureat

the respective temperature as described in Sect. 7.3. As a consequence, the

multiplication rate of (5.2) contains the applied stress but not explicitly the

temperature. However, the latter enters the rate of dislocation annihilation

as climb is involved in this process so that the evolution of the dislocation

density depends strongly on temperature, as will be discussed below.

While cross slip is a prerequisite to dislocation multiplication, localized

sources should operate if cross slip is limited. This is obvious for dislocations

with 1/2111 Burgers vectors in MoSi

,wherethereexistonly{110} planes

as easy slip planes. In the other cases, the situation is not so clear, for example,

in Ti-6 at% Al, where slip and cross slip planes are of diﬀerent type, with both

mechanisms of dislocation generation occurring as demonstrated in Videos

8.4–8.6. The type of dislocation generation certainly inﬂuences the spreading

of slip. Planar slip as it is observed, for example, in MoSi

and Ti–6Al requires

the operation of localized sources. Slip localization is frequently accompanied

with plastic instabilities (Sect. 5.3). Both phenomena are usually discussed on

166 5 Dislocation Kinetics, Work-Hardening, and Recovery

the basis of processes that soften the material, for example, by destroying the

long-range or short-range order (as in Ti–6Al), thus facilitating dislocation

motion in a narrow region. At the same time, the mechanism of dislocation

creation must generate the new dislocations within this narrow region. This

may occur either by the localized Frank–Read sources or by multiplication

at high stresses where cross slip events of small height are suﬃcient for the

emission of a new dislocation.

5.1.3 Dislocation Immobilization and Annihilation

After their formation, the dislocations move through the crystal. If there are

no other obstacles like grain or phase boundaries and if the dislocation den-

sity is low, the dislocations can travel until they reach the crystal surface,

produce a slip step by shifting the two sides of the slip plane against each

other by the Burgers vector, and disappear. Otherwise, the traveling distance

is limited by a number of processes. In suﬃciently large grains or in single

crystals the dislocations may become immobile by being captured in a region

of high internal stress caused by the presence of other dislocations. The basic

mechanism is the formation of dislocation dipoles owing to the mutual attrac-

tion of dislocations with Burgers vectors of opposite sign. As described in

Sect. 3.2.4, dislocations on parallel slip planes have equilibrium positions with

respect to glide. If the applied stress is lower than the critical stress deﬁned by

the dipole opening criterion (3.27), these dislocations cannot pass each other.

0.5 µm

Fig. 5.10. Region of a high density of dislocation dipoles in a deformed MgO

single crystal. Dipoles imaged in inside contrast (i) or outside contrast (o). Single

dislocations (s). Micrograph from the work in [281]

5.1 Dislocation Kinetics 167

According to Tetelman [290], the dislocations with line directions not being

parallel can reorient to arrange themselves parallel and to form the dipole. As

discussed in Sect. 3.2.4, the energy of the dipole is lower than that of the two

isolated dislocations. In contrast to the short dipoles trailed behind high jogs

(e.g., Fig. 5.8), the dipoles formed by the mutual capturing of moving dislo-

cations are mostly very long as demonstrated by the dislocation structure in

a deformed MgO single crystal in Fig. 5.10. The dipoles are imaged as thin

double lines (o) or as thick lines of high contrast (i). In transmission electron

microscopy, the image of a dislocation is shifted with respect to its geometri-

cal projection. Direction and sense of the shift depend on the direction and

sign of the Burgers vector and the imaging g vector. Therefore, the images of

the two dislocations constituting the dipole are shifted in opposite directions.

The contrast appears either inside or outside the geometrical projection of

the dipole. Dipoles of small height imaged in inside contrast (i) often appear

as thick single lines, while dipoles in outside contrast (o) appear as two thin

separated lines.

Further dislocations can be captured in the vicinity of the dipoles. The

static equilibrium situations between an inﬁnitely long edge dislocation dipole

and an approaching individual edge dislocation are investigated in [291]. The

dipole can trap another dislocation in several conﬁgurations. Under stress,

the formed free tripole moves like a single dislocation as the dipole has no

long-range stress ﬁeld. If one dislocation of the dipole is ﬁxed, the approach-

ing single dislocation may decompose the dipole. The dynamics of encounters

between an edge dislocation dipole and a third edge dislocation (tripole con-

ﬁguration) or another widely spaced dipole (quadrupole) was studied in [292],

assuming a linear dependence of the dislocation velocity on the local stress as

in the Peierls mechanism. There are four situations possible.

1. The moving dislocation passes the dipole without changing it.

2. All dislocations form a stable conﬁguration, that is, the moving disloca-

tions are trapped.

3. The dipole is decomposed so that all dislocations become mobile. The

stress necessary for this decomposition with the help of another dislocation

is only half the value of the dipole opening criterion (3.27). Accordingly,

this process is very frequent.

4. The dislocations rearrange to form a new dipole.

As a consequence, dislocation structures of a low density are very dynamic.

Nevertheless, further dislocations are trapped in the regions of an increased

dislocation density, yielding complex dislocation structures with the separa-

tion of regions of a high dislocation density from those of a low one. In these

structures, the total elastic strain energy is lower than that of the same number

of dislocations arranged randomly. The structures limit the traveling distances

of the moving dislocations.

In addition to the 45

◦

dipole conﬁguration of two edge dislocations of

opposite sign, two edge dislocations of equal sign form a stable equilibrium

168 5 Dislocation Kinetics, Work-Hardening, and Recovery

conﬁguration when they arrange themselves along the direction perpendicular

to their Burgers vector, as listed in Table 3.1. Further dislocations on parallel

planes can be trapped to form an irregular tilt boundary. Climb can result in a

regular boundary structure if the temperature is high enough. The formation

of tilt boundaries during recovery is called polygonization.

The kinetics of the mobile dislocation density can be described by com-

bining the formation rate of dislocations (5.2) with an expression for the

immobilization. The basic process of the immobilization is the meeting of two

dislocations. The respective rate is therefore proportional to the square of the

dislocation density [293]

d

−

= −q

ds.

The process coordinate is the slip displacement s as the dislocations meet while

they are traveling. As an elastic process, the temperature does not enter the

immobilization rate. Together with the Orowan equation (3.5), the diﬀerential

equation for the dislocation density may then read as

d =(wτ

∗

 − q

)ds =

(wτ

∗

− q)dγ. (5.3)

In [293], the dependence of the creation rate on the stress is not con-

sidered. The equation can then be integrated to yield a mobile dislocation

density approaching exponentially a steady state value. A linearly increas-

ing total (mobile and immobile) dislocation density is obtained by neglecting

the immobilization rate. The outlined dislocation kinetics is typical of low

temperatures where dislocations cannot disappear.

As the dislocations in the dipole-like conﬁgurations attract each other,

they tend to mutually annihilate. For screw dislocations, this is possible by

cross slip if the attractive force is higher than the glide resistance on the cross

slip plane. Figure 5.11 presents an example of the annihilation of segments of

screw dislocations of opposite Burgers vectors during in situ deformation in

MgO. The annihilated segments are marked by white triangles. The instan-

taneous annihilation of the whole dislocations is prevented by the bowing of

the dislocation segments between localized obstacles on the main slip plane

so that parts of the dislocations cannot orient in pure screw direction. The

ﬁgure shows also the reorientation of the dipole D1 and the disappearance of

the dipole D2 between Fig. 5.11b and c. The latter might have disappeared

by the annihilation of one of its constituting dislocations by another gliding

dislocation, while the remaining dislocation moved away and stopped as dislo-

cation d in Fig. 5.11c. As annihilation by cross slip is restricted to pure screw

dislocations, it is probably not a very frequent process.

Non-screw dislocation dipole conﬁgurations can annihilate only by climb.

The velocity of dislocations of a dipole climbing against each other was brieﬂy

outlined in Sect. 4.10.4. As in other cases of diﬀusion-controlled climb, the

climb velocity is proportional to the self-diﬀusion coeﬃcient, for example,

(4.93). If the climbing velocity along the dislocations of the dipole is not uni-

form, the dipole can be pinched-oﬀ via pipe diﬀusion into small prismatic

5.1 Dislocation Kinetics 169

0.5 µm

(a)

(b)

(c)

(d)

Fig. 5.11. Mutual annihilation of segments of two screw dislocations as well as

annihilation and reorientation of dipoles during in situ deformation of an MgO

single crystal. Micrographs from [79]

dislocation loops. Further climb is then driven by the line tension. TEM

observations of the loop shrinkage during annealing yield information on the

self-diﬀusion coeﬃcient. As there is some similarity to the annealing of dis-

locations, the method will brieﬂy be outlined following Seidman and Baluﬃ

[294].

The prismatic dislocation loop is considered to be of toroidal shape of ring

radius r

loop

and radius of the tube r

. The ﬂux of the point defects emitted

from or absorbed at the surface of the toroid is controlled by diﬀusion into

the lattice having the equilibrium concentration of point defects. The ﬂux per

unit length of the dislocation is then given by

J =



2πr





12D





exp

− 1

