Messerschmidt U. Dislocation Dynamics During Plastic Deformation

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312 8 Metallic Alloys

1 µm

Fig. 8.16. Dislocations moving in a specimen with OR A2 on a plane with a trace

parallel to the interface. From [498]. Copyright (2004) Carl Hanser Verlag, M¨unchen

the original dislocation moves away. Finally, two new dislocations move away. The

last two sequences present dislocations moving away from the crack. In the third

part, the crack shortly appears at the upper edge of the frame. In the last sequence,

the crack is located outside the image below the lower right corner.

To summarize the above, extensive plastic deformation occurs near the

crack tip but also quite far away from it. Many dislocations move on planes

of {112} type with high macroscopic orientation factors. Near the interface,

dislocations also move on slip systems with zero macroscopic orientation fac-

tors, indicating a complicated stress state. New dislocations are created by

the double-cross slip multiplication mechanism far from the crack. However,

many dislocations originate also from regions near the crack tip. It cannot be

decided whether these dislocations are emitted from the crack or multiply in

regions of high stress near the crack tip.

Intermetallic Alloys

Intermetallic alloys are a special class of materials with mechanical proper-

ties that bridge the gap between metals of relatively low yield stress and

good plasticity, and ceramics of high yield stresses even at high temperatures

but strong brittleness at low ones. Many properties of intermetallic alloys

result from their lower crystal symmetry. They increasingly gain structural

applications at high temperatures because of their good strength-to-weight

ratios.

9.1 Introduction

Ordered intermetallic compounds are alloys with two or more metal con-

stituents, the atomic concentrations of which are close to ﬁxed stoichiometric

ratios. These constituents may form long-range order so that the diﬀerent

species of atoms occupy diﬀerently designated lattice sites in typical super-

lattice structures, which are based on the three principal metal structures,

i.e., the f.c.c., b.c.c., and h.c.p. structures. Between the diﬀerent atom species

there may exist strong and also frequently directional bonds. The materials

may have diﬀerent degrees of atomic disorder resulting from low ordering

energies or from deviations from the stoichiometric compositions. These devi-

ations are realized by vacancies in speciﬁc sublattices or by antisite defects,

i.e., by atoms occupying the sites of other constituents. General properties

of intermetallic compounds are summarized, e.g., by Sauthoﬀ [501] and by

Westbrook and Fleischer [502].

In intermetallic compounds, the dislocations have complete Burgers vec-

tors which are mostly multiples of the Burgers vectors in the corresponding

simple metal structures. These superdislocations dissociate into superpar-

tial dislocations including planar faults like antiphase boundaries (APBs)

and diﬀerent kinds of stacking faults. The dislocations may have compli-

cated three-dimensional core structures, which impede the dislocation motion.

The properties of dislocations and their relation to the characteristics of

314 9 Intermetallic Alloys

macroscopic deformation were summarized by Yamaguchi and Umakoshi

[503].

Many intermetallic alloys show a ﬂow stress anomaly, i.e., an anomalous

increase of the ﬂow stress with increasing temperature within a certain tem-

perature range, mentioned earlier in Sect. 4.11 (Fig. 4.40). The anomaly is a

desired property for structural applications since it retains a high strength

up to high temperatures. The physical origins may be manifold and are fre-

quently connected with transitions of the dislocation cores into an immobile

state. On the other hand, the occurrence of the anomaly is often quite unspe-

ciﬁc of the particular material. It is observed also at deformation by ordinary

dislocations with simple cores. In these cases, the anomaly may be caused by

diﬀusion processes as discussed in Sect. 4.11.

The basic features of the structure of intermetallics will be introduced on

materials of the A

B composition like Ni

Al. These alloys frequently crystal-

lize in the L1

structure shown in Fig. 9.1a. The B atoms occupy the corners

of the underlying f.c.c. lattice, whereas the A atoms are situated in the face

centered positions. The arrangement of the atoms on the (111) slip plane is

illustrated in Fig. 9.1b. The complete Burgers vector is of type 110, e.g.,

from the B atom at 0 to the B atom at β. It is twice as long as the Burgers

vector in the f.c.c. structure (see, e.g., [504]). The dislocations can dissociate

into two superpartial dislocations with Burgers vectors of 1/2110, e.g., one

from 0 to α and the other one from α to β. The shift of the a layer and all

above it by the superpartial Burgers vector b

=0α =1/2[

110] creates an

antiphase boundary (APB) as a planar fault. The fault may be annihilated by

the second superpartial with the Burgers vector αβ. Superlattice dislocations

enclosing an APB were ﬁrst observed in [505] and have been mentioned before

for cutting the particles with the L1

structure in an Al–Li alloy (Sect. 8.1.3).

[001]

[010]

[100]

(a)

[–210]

[–101]

[11–2]

[01–1]

[–110]

[–12–1]

(b)

Fig. 9.1. Unit cell (a) and atomic arrangement on the slip plane (b)ofanA

compound of the L1

structure. Three layers of atoms a, b, c are stacked in the

...abcabc... sequence and are drawn by circles of decreasing size

9.1 Introduction 315

Other possible shift vectors on the octahedral plane are b

=0γ =1/3[

1],

which creates a so-called superlattice intrinsic stacking fault (SISF), and

=0δ =1/6[

210] producing a complex stacking fault (CSF). The SISF

on the {111} plane causes an ...abab... (h.c.p.) stacking not violating the

nearest neighbor relations. It corresponds to a usual stacking fault in the f.c.c.

structure. The CSF also produces an h.c.p. stacking, however, violating the

nearest neighbor relations. A method of studying the stability of the diﬀerent

planar faults is the construction of the so-called γ surface developed by V.

Vitek for b.c.c. metals [506]. A planar fault is created by cutting the crystal

along a plane and shifting both parts with respect to each other by a fault

vector f. The fault energy γ is determined by suitable atomistic methods

(pair potential, embedded atom, ﬁrst principles, or other methods) and plot-

ted against f . The minima in this plot, the γ surface, are the positions of

the stable fault vectors. Calculations of the γ surface for the L1

structure

with diﬀerent central force potentials [507] showed that ABPs with the above

fault vector b

arealwaysstableon{001} planes since the nearest neighbor

relations are not violated. APBs on {111} planes may be stable or unstable

depending on the applied potential. The fault vector may have a component

out of the fault plane. The SISFs are always stable on the {111} plane. Similar

planar faults exist also in the other crystal structures of intermetallics. The-

oretical calculations of the γ surfaces and the core structure of intermetallic

alloys are reviewed by Vitek and Paidar [508].

The diﬀerent mechanisms explaining the ﬂow stress anomaly in inter-

metallics can be classiﬁed into three groups. The details of the models will be

described in the sections of the particular materials:

1. Formation of sessile dislocation locks due to particular nonplanar core

conﬁgurations. These mechanisms are speciﬁc for particular crystal and

dislocation core structures of intermetallics. Such sessile cores can result

from cross slip of the leading superpartial from the {111} slip plane in the

structure onto a {100} cross slip plane to reduce the total energy of the

dislocation. The cross slip is supposed to be thermally activated so that

the rate of formation of the locks increases with increasing temperature.

The locked conﬁgurations may be unlocked by the formation of double-

kinks which smoothly spread along straight dislocations as for the simple

Peierls mechanism (Sect. 4.2.2) or by unlocking long segments leading to

a jerky motion (locking–unlocking mechanism, Sect. 4.4).

2. Cross slip processes not speciﬁc for the particular dislocation core struc-

ture. Segments of a screw dislocation may cross slip onto another slip

plane and bow out on this plane so that the dislocation contains segments

bowing out on diﬀerent planes. The cusps between the segments are then

not glissile in forward direction, they pin the dislocation. A variant of this

model is the usual double-cross slip mechanism described in Sect. 5.1.1,

where the jogs formed by double-cross slip are not glissile in forward

316 9 Intermetallic Alloys

direction of the whole screw dislocation. The anomaly again results from

higher cross slip frequencies at higher temperatures.

3. Various mechanisms involving point defects and diﬀusion. In the most

simple case, thermal vacancies with their concentration increasing with

increasing temperature are considered ﬁxed localized obstacles to the dis-

location motion. Besides, diﬀusing intrinsic and extrinsic point defects

may form point defect atmospheres around the dislocations, which cause

a diﬀusional drag as described in Sect. 4.11. Finally, dislocation cores may

decompose into components with nonplanar Burgers vectors. The whole

dislocation can then move only by the combination of glide and conserva-

tive climb, i.e., a point defect ﬂow from one component dislocation to the

other.

All these mechanisms explain the occurrence of a ﬂow stress anomaly.

However, they have diﬀerent implications on the shape of the dislocations

and their kinematic behavior, the stability and the activation parameters of

deformation, and on the orientation dependence of the ﬂow stress. Although

a wealth of experimental data has been collected on the deformation of the

diﬀerent intermetallic materials, an unequivocal interpretation of these data

is frequently not yet possible. The following materials are discussed mainly

with respect to the ﬂow stress anomaly.

9.2 Ni

The L1

ordered γ



Al phase is the strong constituent of the γ/γ



nickel

base superalloys. These alloys combine high strength at high temperatures

with a good oxidation resistance. In single crystal form, they are employed,

e.g., as turbine blades in the hot parts of aero jet engines. For a review, see

[509]. The γ



phase has been studied for about 50 years. It precipitates in the

form of cubes regularly arranged in a disordered γ phase matrix. The matrix

forms narrow channels between the precipitates. Plastic deformation starts in

these soft channels. The dislocations with 1/2110 Burgers vectors become

superpartial dislocations when they enter the hard γ



phase. Ni

Al shows a

ﬂow stress anomaly with a peak temperature of about 750

◦

C, which is respon-

sible for the high strength of the Ni base superalloys at high temperatures.

The anomaly is connected with a high work-hardening rate and a low strain

rate sensitivity of the ﬂow stress.

9.2.1 Microscopic Observations and Dislocation Dynamics

Al may be considered a prototype material of intermetallics. Its dislocation

properties are reviewed in several papers (e.g., [510]) and will therefore only

brieﬂy be reported here. The crystal structure and possible planar faults were

described above. The slip system dominating the anomaly range of the ﬂow

9.2 Ni

Al 317

stress is {111}110. Several dissociation schemes have been discussed. Those

most important for the octahedral plane are (e.g., [503])

[

110] = 1/2[

110]+APB+1/2[

110] (9.1)

=1/6[

211]+CSF+1/6[

1] + APB

+1/6[

211]+CSF+1/6[

1]. (9.2)

In [511], the fault energies were determined by weak-beam TEM. As men-

tioned earlier, the APB energy may be lower on the cube plane than on the

octahedral one so that the APB extends also on this plane, depending on

the ratio between the energies on both planes. This will be discussed in more

detail in the next section. Dislocation conﬁgurations have been investigated in

many studies mainly by TEM, which is not described here. The dislocations

are then in a static conﬁguration which need not be characteristic of mobile

dislocations.

In situ experiments in a TEM were performed by Mol´enat and Caillard

[512, 513]. From this work, a movie sequence of dislocation motion at room

temperature is presented in the following video.

Video 9.1. APB jumps in Ni

Al-0.25at%Hf at room temperature (from the work

in [512]). Courtesy of D. Caillard: The weak-beam images of the dislocations are

straight in screw orientation. The dislocations are dissociated into superpartials

connected by an APB. Further dissociation of the superpartials cannot be resolved

during the in situ experiments. Within the time resolution of the video recordings,

the dislocations move in jumps. The width of the jumps can be quite large, e.g., a new

dislocation appears suddenly in the upper part of the frame. Frequently, however,

the dislocations jump over distances comparable with the width of the dislocations.

This is shown in Fig. 9.2. The contrast feature marked by a triangle may be taken

as a reference point. Such jumps over the distance of the dissociation width are

called APB jumps. Only part of the upper dislocation in the movie undergoes an

APB jump. The moved and un-moved parts are connected by macrokinks, as will be

discussed in the next section. A detailed analysis shows that the dislocations in the

mobile conﬁguration extend mainly on the octahedral slip plane. Only dislocations

at rest are dissociated on the cube cross slip plane. These features have been revealed

only by in situ experiments. It was only later, that this mechanism was observed by

conventional TEM, too [514].

A similar behavior is observed up to about 300

◦

C [513]. Above this tem-

perature up to the peak temperature, dislocation motion on {111} planes

during in situ experiments takes place in avalanches with large slip distances

on many slip planes. After motion, the dislocations are locked by dissociation

on the cube plane. Under shear stress, they may bow out on this plane.

Al is probably the only intermetallic material in which dislocation

velocities have been measured as a function of stress and temperature.

Figure 9.3 presents the results of a study by Nadgorny and Iunin [515] using

the stress pulse-double etching method (Sect. 2.2). The dislocations were cre-

ated by a scratch. The velocities were measured on groups of dislocations

318 9 Intermetallic Alloys

(b)(a)

Fig. 9.2. Two stages of an APB jump in Ni

Al-0.25at%Hf at room temperature.

From the work in [512]. Courtesy of D. Caillard

Fig. 9.3. Dependence of the dislocation velocity on the resolved shear stress τ in

Ni-22.9at%Al single crystals at diﬀerent temperatures. Data from [515]

having moved out of the scratch. The stress necessary to reach a certain

dislocation velocity strongly increases with increasing temperature, which cor-

responds to the ﬂow stress anomaly. At room temperature, the stress to attain

avelocityof3nms

−1

is equal to the macroscopic ﬂow stress. At high tem-

peratures, it is lower, probably due to an increased athermal contribution of

long-range dislocation interactions. The slope of the curves corresponds to the

stress exponent m according to (4.11). The values of m are 4.2, 9.6, 9.5, and

31 for the temperatures of 81, 295, 673, and 873 K. The low values are con-

sistent with the dislocation velocity being controlled by a thermally activated

process. The high value of 31 at 873 K agrees with the usually observed low

strain rate sensitivity in the anomaly range. These measurements are the only

direct proof that the ﬂow stress anomaly is caused by a reduction of the dislo-

cation mobility and not by collective dislocation processes like work-hardening

although the anomaly is mostly connected with high work-hardening rates.

9.2 Ni

Al 319

{111}

{100}

APB

(a) (b) (c)

Fig. 9.4. Locking of a dislocation in the L1

structure by cross slip (Kear–Wilsdorf

lock)

9.2.2 Models of the Flow Stress Anomaly

The basic model of explaining a ﬂow stress anomaly by special properties of

the dislocation cores in intermetallic materials is the formation of the so-called

Kear–Wilsdorf locks after the paper by Kear and Wilsdorf [516]. As shown

in (9.1), the superdislocation in the L1

structure can dissociate into two

superpartials enclosing an APB. The superpartials can further split including

CSFs as outlined in Fig. 9.4a. One superpartial may cross slip onto the {100}

cross slip plane (Fig. 9.4b) if the APB energy is lower on this plane. The

cross-slipped superpartial may then dissociate again on a plane diﬀerent from

{100} making the whole dislocation immobile. A ﬁrst quantitative theory

of the ﬂow stress of intermetallics with the L1

structure including the ﬂow

stress increase with rising temperature in the anomaly range and the following

high-temperature decrease was given by Takeuchi and Kuramoto [517]. The

dislocations are assumed to form localized Kear–Wilsdorf locks by thermally

activated cross slip from the {111} glide plane onto the {100} cross slip plane.

The thermal activation is assisted by the shear stress on the cross slip plane

so that an equation like (4.6) holds for the rate of activation. For the dynamic

breakaway from these locked conﬁgurations, the shear stress on the glide plane

has to exceed a critical value, and for the steady state motion, a condition has

to be fulﬁlled similar to the Friedel criterion for localized obstacles (Sect. 4.5.1)

that one new obstacle has to be formed after the dislocation has moved to

the critical conﬁguration for breakaway. The ﬂight motion follows the viscous

law (4.74). Because of the thermally activated nature of the formation of

Kear–Wilsdorf locks by cross slip, the ﬂow stress increases with increasing

temperature. Above the peak temperature, strongly temperature depending

slip on the cross slip plane leads to a rapidly decreasing ﬂow stress.

A considerable extension of the theory was given by Paidar, Pope and

Vitek [518]. It is usually quoted as the PPV model. In this theory, the forma-

tion of a kink pair of the leading superpartial on the {100} cross slip plane

is considered as depicted in Fig. 9.5. The energy for the formation of the

double-kink is written as

H = W +2Ch − ΔEL − τ

bLh −Mh

/2L.

Here, the ﬁrst term is the energy of the two constrictions which have to be

formed on the leading superpartial before cross slip can start. The second

320 9 Intermetallic Alloys

{111}

{100}

(a)

(b)

Fig. 9.5. Formation of a kink pair in the L1

structure from the {111} glide plane

onto the {100} cross slip plane (a) and cross slip back to a parallel {111} plane (b).

The dissociation of the superpartials into Shockley partials is not shown. After [520]

term is the energy of the kink pair on the cross slip plane with C being the

speciﬁc kink energy per length, and h is the kink height (or width of the

cross-slipped segment). The third term describes the gain in energy due to

the change of the APB from the octahedral slip plane onto the cube cross slip

plane. It is proportional to the length L of the cross-slipped segment. ΔE is

positive only if γ

<γ

√

3. γ

and γ

are the APB energies on the {100} and

{111} planes. The fourth term is the work done by the external stress τ

the cross slip plane. Finally, the last term represents the interaction energy

between the two kinks in a screw dislocation (4.23). The activation energy

for the cross slip process is given for the saddle point conﬁguration where the

length L takes a critical value. The kink height is either b/2orb. In the PPV

model, the activation energy for cross slip, i.e., that for locking, is controlled

by four eﬀects: the diﬀerence in the APB energies γ

and γ

, the shear stress

on the cross slip plane, the elastic energy diﬀerence between the stress induced

compression or dilatation of the superpartial pair before a jump and the equi-

librium conﬁguration afterwards, and ﬁnally, the nature of the dissociation

of the superpartials on the octahedral planes. For the unlocking process, the

same assumptions are made as for the model of Takeuchi and Kuramoto [517].

The PPV theory explains the occurrence of a ﬂow stress anomaly as well as

its orientation dependence and a tension–compression asymmetry, which all

have been observed experimentally.

The models described so far do not consider certain microstructural

observations, in particular the occurrence of the APB jumps in the lower

temperature range of the ﬂow stress anomaly (Video 9.1). D. Caillard and

V. Paidar have extended the theory to include these features [519, 520] by

assuming that the locking cross slip process is of short range and occurs only

over widths of b or a few b to form incomplete Kear–Wilsdorf locks. Unlock-

ing takes place by a second cross slip back to a plane parallel to the original

9.2 Ni

Al 321

{111}

{100}

CSF

APB

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

Fig. 9.6. Dislocation jumps over variable distances (line A) and APB jumps (line

B) in intermetallics with the L1

structure. L locking, UL unlocking. After [519]

octahedral slip plane, similar to the double-cross slip mechanism for disloca-

tion multiplication (Sect. 5.1.1). As illustrated in Fig. 9.6, a dislocation in the

mobile conﬁguration (a) in line A locks (L) itself by cross slip of the leading

superpartial onto a cube plane (b). In (c), it is still locked by the trailing

superpartial. When the latter removes the APB on the cube plane by glid-

ing to the plane of the leading superpartial, the dislocation is unlocked (UL)

and the mobile conﬁguration is restored (d). The dislocation can then glide

over relatively long variable distances. This process is one case of the locking–

unlocking mechanisms brieﬂy described in Sect. 4.4. In line B of Fig. 9.6, the

sequence starts with a dislocation locked at both superpartials (a). The dislo-

cation can unlock to take the same semi-locked conﬁguration (b) as in line A.

If the leading superpartial in (c) cross slips before the trailing one is unlocked,

the whole dislocation is completely blocked again (d). The slip distance now

corresponds to the width of the APB in the locked conﬁguration. Since the

cross slip heights are small in both cases, the macroscopic plane of motion

does not deviate remarkably from the octahedral glide plane. With increasing

temperature, locking becomes more complete, i.e., the dislocations spread on

the cross slip plane. These complete Kear–Wilsdorf locks unlock only at high

temperatures.

In [519], all forces acting on the long and straight superpartial dislocations

are calculated, similar to [521,522]. The ranges of the existence of incomplete

Kear–Wilsdorf locks in plots of the normalized stress on the octahedral glide

plane on the cross slip height are then established for a ﬁxed value of γ

/γ

0.8 and diﬀerent values of stress on the cross slip plane. The extension of the

model [520] takes into account the thermally activated nature of the cross

slip events. It also distinguishes between a process similar to the kink pair

formation with a cross slip height h = b as shown in Fig. 9.5a with a low

activation energy to form incomplete Kear–Wilsdorf locks, and cross slip over

larger distances, which is controlled by a critical bow out in the cross slip

plane. This process requires a higher activation energy and leads to incomplete

Kear–Wilsdorf locks of diﬀerent strengths and to complete locks. In both cases,