Messerschmidt U. Dislocation Dynamics During Plastic Deformation

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332 9 Intermetallic Alloys

1µm

[1-10]

Fig. 9.14. Superdislocations in γ-TiAl under load which had moved at 655

◦

g = [001] diﬀraction vector. From the work in [286]

Fig. 9.15. Temperature dependence of the yield stress of the intermetallic alloys γ-

Ti-52at%Al, NiAl and NiAl-0.2at%Ta. The NiAl single crystals are deformed along

the soft 421 orientation. Data from [535]

a single relaxation curve (see Sect. 2.1). These relaxation curves cover more

than three orders of magnitude of the strain rate. r is low at room temperature

as listed in Table 9.1. At and above 400

◦

C, the relaxation curves show the

typical “inverse” curvature with a low strain rate sensitivity at high stress (or

9.3 γ-TiAl 333

Table 9.1. Dependence of the strain rate sensitivity r and activation volume V in

γ-TiAl on the temperature

T (

◦

C) r (MPa) V (nm

)

21 0.913.5

400 2.27 12.3

570 4.28.3

630 7.55 4.95

700 12.73.17

Fig. 9.16. Dependence of the strain rate sensitivity of the stress in γ-Ti-52at%Al

on the logarithm of the strain rate with the temperature as parameter. Data from

[535]

strain rate), and with a higher strain rate sensitivity at lower stresses, like the

curve R1 in Fig. 8.2b measured on Al–Zn–Mg. Strain rate sensitivities were

determined at diﬀerent points along the relaxation curves and then plotted

versus the actual strain rate. The latter was obtained from the stress rate

using the stiﬀness of the sample and ﬁxtures during unloading at the end of

the experiments according to (2.7). Data from experiments at diﬀerent tem-

peratures are plotted in Fig. 9.16. Table 9.1 presents the values at ˙ε =10

−5

−1

together with the corresponding activation volumes V from (4.9), based on

an average orientation factor of m

=1/3. The activation volumes are con-

stant up to 400

◦

C before they decrease. At 400

◦

C, the strain rate sensitivity

decreases only weakly with increasing strain rate. With increasing tempera-

ture, r increases, but its decrease with increasing strain rate becomes very

pronounced. In general, only a minor part of the total ﬂow stress (about 30%

at 630

◦

C) relaxes during usual relaxation times (about 10 min).

9.3.4 Deformation Mechanisms

Though the studied single-phase γ-TiAl shows only a slightly increasing

ﬂow stress and a decreasing activation volume over the whole temperature

range investigated, the dislocation processes diﬀer greatly between room

334 9 Intermetallic Alloys

temperature and high temperatures. Nevertheless, the microstructure is always

characterized by the presence of ordinary dislocations, superdislocations and

microtwins with a clear dominance of ordinary dislocations. The diﬀerence in

the role of the three types of dislocations with respect to the results in [536]

may be due to the lower aluminium content of the present material.

Room Temperature up to About 430

◦

The essential experimental observations in the low-temperature range are as

follows:

• Ordinary dislocations bowing out between obstacles about 100 nm distant

dominate the deformation.

• The cusps in the dislocation line forming at the obstacles are aligned not

exactly along the screw orientation but are spread on the slip plane.

• Neighboring segments of ordinary dislocations bow out on the same {111}

plane, or on parallel planes.

• Under stress, the bowing decreases with increasing temperature.

• Superdislocations with 101 Burgers vectors show the same bowed-out

shape as the ordinary ones do.

• All types of dislocations show the same kind of jerky motion.

Ordinary dislocations dominate the deformation in the described experi-

ments at all temperatures. In contrast to [537] but in agreement with [538],

they have a high mobility. The theory predicts that ordinary dislocations have

extended cores leading to a low mobility [539–541]. Experimental observations,

however, show that the cores of ordinary dislocations should be rather compact

[542]. Nevertheless, theoretical estimates suggest a high Peierls stress (e.g.,

[543]), which agrees with the observation that dislocation segments bowing

out under stress relax only little after unloading.

A number of transmission electron microscope studies have been carried

out to assess the contributions of the diﬀerent dislocation types at diﬀerent

temperatures. While some authors mainly observe superdislocations at room

temperature (e.g., [537]) others ﬁnd a dominance of ordinary dislocations at

all temperatures (e.g., [544]). Apparently, the activation of the diﬀerent types

of dislocations depends on the concentration of oxygen impurities within the

specimens [544–546] as well as on the content of aluminium [536]. In the

experiments described above, superdislocations do not play an important role.

Thus, the following discussion is based mainly on the properties of ordinary

dislocations.

The literature oﬀers three models to explain the curly shape of ordinary

dislocations, the ﬁrst two of which are also thought to cause the ﬂow stress

anomaly at higher temperatures:

1. Louchet and Viguier [547] and Viguier et al. [548] assume that neighbor-

ing segments of ordinary screw dislocations cross slip on diﬀerent {111}

planes and bow out on them as schematically drawn in line a of Fig. 9.17.

9.3 γ-TiAl 335

Fig. 9.17. Models of pinning by cross slip in TiAl. Line a: Cross slip of individual

segments on diﬀerent planes according to [547, 548]. Line b: Double-cross slip as

described in Sect. 5.1.1, Fig. 5.5

The points linking these segments are aligned almost exactly in screw ori-

entation. They cannot glide together with the dislocations so that they

form obstacles. The dislocations move very jerkily by pushing the pin-

ning links along the dislocations in the direction of the Burgers vector to

form long mobile segments, which spread very quickly. This mechanism is

also called the local pinning unlocking (LPU) model. The present obser-

vations are not in accordance with this model as the obstacles are aligned

not exactly along the screw orientation. Besides, obstacles occur along all

types of dislocations, including edge dislocations. While jogs in edge dis-

locations may impede the dislocation motion by their lattice friction, too,

leading to bowed-out segments, edge dislocations cannot cross slip, which

is a prerequisite to the formation of pinning agents according to the LPU

model. Besides, stereo images do not show neighboring segments bowing

out on diﬀerent crystallographic planes but mostly on parallel ones.

2. Sriram et al. [549] carefully investigated the spatial arrangement of the

ordinary dislocations by post-mortem electron microscopy. They found

the individual segments bowing out on parallel planes and the cusps not

strictly aligned in screw direction as illustrated by line b in Fig. 9.17.

The authors assume that this structure of the dislocations is formed by

the double-cross slip mechanism described in Sect. 5.1.1, i.e., all cusps in

the dislocations correspond to jogs with a spectrum of heights ranging

from the atomic height up to a maximum height which is determined

by the dipole opening criterion. These experimental observations are in

accordance with most of the features of the microstructure of the resting

dislocations at low temperatures. Deep cusps or the trailing of dipoles sug-

gest that the ordinary screw dislocations have many jogs. However, cusps

were also observed in mixed and edge dislocations as proved in Fig. 9.9.

As discussed earlier, edge dislocations cannot cross slip to form these jogs.

The kinematic behavior of dislocations presented in the Video 9.2 and

in Fig. 9.11 does not agree with jogs being the dominating obstacles on

screw dislocations. The jogs can glide along the direction of the Burgers

336 9 Intermetallic Alloys

vector. This motion leads to long segments with very large bow-outs,

before the conﬁguration gets unstable. This mechanism was identiﬁed in

in situ straining experiments on NiAl single crystals at room temperature

[287] and will be described in Sect. 9.4. Such long segments have not

been observed in TiAl. Thus, while the double-cross slip mechanism may

explain most of the features of the resting dislocations as also observed

in post-mortem studies, it fails to explain the kinematic behavior of

dislocations in the in situ experiments at room temperature.

3. Almost all experimental results can be interpreted consistently if it is

assumed that during the deformation between room temperature and

about 430

◦

C, in addition to deep cusps at high jogs, the weak cusps

in the dislocations result from an interaction between dislocations and

extrinsic spatially localized obstacles following the theory described in

Sect. 4.5.1, i.e., probably small precipitates. The main arguments are the

following. The cusps in the resting ordinary dislocations are lined up not

exactly along the screw orientation but are distributed on the slip plane

(Fig. 9.8). The cusps appear in all types of dislocations, also in edge and

mixed ordinary dislocations (Fig. 9.9), superdislocations (Fig. 9.10) and in

twinning partial dislocations (Video 9.5, see also [532]). The latter disloca-

tions are not able to cross slip on {111} planes to form jogs. The obstacle

distance l is independent of the character and type of dislocations and of

the temperature, and the activation volume is independent of the temper-

ature too (Table 9.1). The dynamic behavior of dislocations is compatible

with precipitation hardening.

Some quantitative estimates are following regarding the precipitation hard-

ening mechanism. It is argued in [286] that the precipitates are small oxide

particles about 4 nm wide. Applying Friedel statistics, the activation volume V

measured in macroscopic tests is related to the obstacle distance l and the size

by (4.55) and (4.4). Using l ≈ 100 nm and V =13.5nm

from above, the acti-

vation distance Δd turns out to be about 0.7 nm or 2.5 b. This ﬁts well the

estimated obstacle width of 4 nm. The small value of Δd indicates that the

equilibrium position of the dislocations waiting for thermal activation is near

the tip of the interaction potential, or that the stress is not much smaller than

the athermal ﬂow stress of the obstacle array. This corresponds to the kine-

matic dislocation behavior partly with jumps over larger distances (Fig. 4.18).

The stress τ

∗

≈ 40 MPa estimated above from the dislocation bowing can

be compared with the macroscopic ﬂow stress in Fig. 9.15. It is important to

note that the bowing of the dislocation segments only slightly relaxes after

the specimens are unloaded. This indicates that a friction stress τ

acts on the

dislocations, which prevents the relaxation and which is only slightly smaller

than τ

∗

. Friction stresses were calculated for ordinary dislocations, e.g., in

[541] using the embedded-atom method. These stresses are high for screw

dislocations, intermediate for 60

◦

dislocations, and low for 30

◦

and edge dis-

locations. This may be a reason why edge dislocations are more mobile than

screw dislocations so that the latter dominate the dislocation structures.

9.3 γ-TiAl 337

In addition to the stress components τ

∗

and τ

, a long-range internal stress

component τ

contributes to the ﬂow stress. An estimation via the Taylor

hardening model (5.11) and an intermediate dislocation density of  =2×

−2

, quoted earlier, yields τ

≈ 44 MPa. Thus, the total resolved yield

stress estimated from the microstructure

= τ

∗

+ τ

≈ 2τ

∗

+ τ

will be about 125 MPa. With the stress value from Fig. 9.15, the macroscopic

resolved yield stress τ

= m

is only about 60 MPa. Thus, the stress com-

ponents resulting from the dislocations bowing out between obstacles, the

friction stress and the long-range athermal stress overestimate the macroscopic

yield stress. There may be several reasons, for instance using an incorrect line

tension in estimating τ

∗

(see the discussion in Sect. 5.2.3). With rising tem-

perature, thermal activation becomes more and more important. This leads

to a decreasing contribution of τ

∗

to the ﬂow stress and, accordingly, to a

weaker curvature of the dislocation segments between the precipitates, as

observed in Fig. 9.12. Finally, the precipitates are overcome spontaneously

and no longer act as obstacles. A very similar behavior of dislocations was

observed in two-phase Ti–Al materials [531].

Flow Stress Anomaly

Several attempts have been made to explain the ﬂow stress anomaly of TiAl.

As reviewed by Vitek [550], the core structure remarkably inﬂuences the prop-

erties of dislocations in ordered noncubic phases. First explanations of the ﬂow

stress anomaly in TiAl were made on the basis of 101 superdislocations by

Hug et al. [551] in analogy to the Kear–Wilsdorf locks in L1

alloys [516].

The energy of the planar faults on the {010} plane is lower than on the {111}

plane. Therefore, dissociated 101 superdislocations may cross slip from the

{111} slip plane to the {010} cross slip plane where they become locked.

In a more general way, the cores of the dislocations may either extend on

their slip plane resulting in a glissile conﬁguration, or they may assume a

three-dimensional structure leading to a low mobility (e.g., [539,540]). While,

at room temperature, superdislocations with 101 Burgers vectors are ﬂex-

ible under load and not locked (Fig. 9.10), at high temperatures they show

straight segments in screw orientation, which are connected by superkinks as

in Fig. 9.14, in accordance with the theory. However, the motion of superdislo-

cations has never been recorded in in situ experiments. Although the diﬀerent

behavior of superdislocations at room temperature and at high temperatures

may explain the ﬂow stress anomaly, their small number suggests that they

do not considerably inﬂuence the high-temperature deformation under the

conditions of the present in situ experiments. Consequently, the ﬂow stress

anomaly has to be explained on the basis of ordinary dislocations only.

The ﬁrst two models by Louchet and Viguier [547] and Sriram et al. [549]

introduced in the foregoing section to interpret the curly shape of ordinary

338 9 Intermetallic Alloys

dislocations in TiAl at room temperature were also used to explain the ﬂow

stress anomaly. Both models assume that cross slip leading to the forma-

tion of pinning centers is thermally activated so that the number of pinning

points increases with increasing temperature. However, above about 430

◦

the dominating ordinary dislocations no longer bow out between localized

obstacles. In the in situ experiments, the ordinary dislocations are created

instantaneously in an avalanche, which had already been observed in earlier

in situ experiments, as reviewed in [552]. If they move again, they are smoothly

bent, moving in a viscous way as shown in Fig. 9.13 and Video 9.6, similarly

to NiAl described in Sect. 9.4. This had not been observed before in TiAl. It

clearly contradicts the cross slip models of the ﬂow stress anomaly. First, the

dislocation segments no longer show the curly shape, and second, the disloca-

tions move continuously, but not in the proposed pinning–unzipping mode. As

discussed earlier, the dislocations are not conﬁned to their slip planes, indi-

cating that climb is involved. Smoothly curved dislocations not lying on a slip

plane were recently observed also in a post-mortem study [553]. This view is

supported by the formation of helical dislocation structures in two-phase Ti–

Al during in situ heating experiments [554]. Climb requires lattice diﬀusion

of both atomic species. The formation and migration energies of vacancies in

TiAl are between 1 and 1.6 eV [555,556]. Thus, these vacancies are present and

mobile in the temperature range around the ﬂow stress maximum. However,

climb usually facilitates dislocation motion and results in recovery so that it

does not induce additional friction.

Since the low-temperature mechanism of precipitation hardening ceases to

operate above 575

◦

C, an additional process must cause the increase or at least

the constancy of the ﬂow stress in the range of the anomaly, associated with

the viscous type of dislocation motion. Viscous motion at high temperatures is

not consistent with simple models of a superposition of long- and short-range

obstacles to dislocation motion, as outlined in Sect. 5.2.2 and Fig. 5.20. In

the high-temperature (athermal) case, the dislocation motion becomes very

jerky. It is a major aim of the intermetallics chapter of this book to stress

that diﬀusion-controlled processes may give rise to additional friction imped-

ing the motion of dislocations in intermetallic alloys. Locking mechanisms

and diﬀusion-controlled processes are compared in [557]. These diﬀusion pro-

cesses may have quite diﬀerent origins as will be discussed in the present and

following sections. γ-TiAl probably is a relatively simple case where point

defect atmospheres form around the dislocations as described in Sect. 4.11.

The atmosphere formation may be superimposed with climb, which follows

from the dislocations not being restricted to their glide planes at high temper-

atures. The atmospheres around dislocations may form from solute impurities

as proposed in [552] to cause strain ageing eﬀects. Such eﬀects have been

observed between 150 and 350

◦

C [558], i.e., at temperatures where the pre-

cipitation hardening ceases to contribute to the ﬂow stress. The amplitude of

strain ageing increases when the Al content decreases below 50%. It is there-

fore concluded that the responsible point defects are antisite defects which

9.3 γ-TiAl 339

have noncentrosymmetrical stress ﬁelds. They may associate with vacancies

which change their position by diﬀusional jumps producing an induced Snoek

eﬀect. In range B of Fig. 4.39, the strain ageing takes place. It is associated

with a negative strain rate sensitivity. A negative steady-state strain rate sen-

sitivity (see Fig. 2.2) was observed at 350

◦

C [559]. The maximum of the stress

contribution from the atmospheres is located left of range B, i.e., at a higher

temperature.

By means of the theory of the Cottrell eﬀect in Sect. 4.11 it was estimated

from (4.100) that several hundred p.p.m. of the diﬀusing defects are necessary

to cause a remarkable contribution to the ﬂow stress. The concentration of the

antisite defects is in the percent range and the concentration of vacancies is

also high at temperatures of the anomaly. Applying the same theory, the strain

rate ˙γ

at the maximum stress can be related to the diﬀusion coeﬃcient D of

the diﬀusing defects by (4.99). Using appropriate data of ˙γ =10

−5

−1

, 

2 ×10

−2

, and the temperature of the ﬂow stress maximum of T = 730

◦

the diﬀusion coeﬃcient of the defects moving in the atmospheres should be

about D =2.4 × 10

−19

−1

. This value ﬁts very well the extrapolated

diﬀusion data of Ti in TiAl [556,560]. It is higher than the diﬀusion coeﬃcient

of Al [556]. On a semi-quantitative level, the present estimates show that

intrinsic point defects in TiAl causing strain ageing eﬀects may considerably

contribute to the ﬂow stress anomaly.

The assumption that point defect atmospheres control the dislocation

mobility in the range of the ﬂow stress anomaly is consistent with the inverse

dependence of the strain rate sensitivity r on the strain rate ˙ε in Fig. 9.16.

The data belong to the stable range A in Fig. 4.39. The viscous motion of the

dislocations in the in situ experiments corresponds to this range of positive

strain rate sensitivity. At high dislocation velocities, the strain rate sensitivity

becomes negative in range B leading to plastic instabilities, which may occur

in TiAl at intermediate temperatures (e.g., [561]). In the in situ experiments,

the instantaneous formation and motion of ordinary dislocations during the

ﬁrst loading belong to range B. It will be discussed in the following sections

that in the respective materials, the occurrence of a ﬂow stress anomaly is

always accompanied by the inverse dependence of the strain rate sensitivity

on the strain rate or stress.

As brieﬂy outlined in Sect. 9.3.3, only a small part of the stress relaxes

during the stress relaxation tests. This is a qualitative indication that a

remarkable part of the ﬂow stress is of athermal nature, which seems to

contradict the view of the dislocation motion being controlled by the thermally

activated processes of point defect atmospheres. However, it was pointed out

already by Hirth and Lothe [12] that, according to the Orowan equation (3.5),

a reduced dislocation mobility increases the mobile dislocation density 

which, in turn, results in an increased athermal component of the ﬂow stress.

Unfortunately, to the authors’ knowledge, there are no representative data

available of dislocation densities in TiAl and many other intermetallic alloys

340 9 Intermetallic Alloys

in the diﬀerent temperature ranges to estimate the temperature dependence

of the athermal stress component.

The in situ experiments in the intermediate temperature range demon-

strate that between about 430 and 600

◦

C the low-temperature mechanism

of the thermally activated overcoming of obstacles changes into the high-

temperature mechanism involving lattice diﬀusion. There is probably a wide

range when both mechanisms are superimposed.

9.3.5 Summary

In γ-TiAl at room temperature, all types of dislocations, i.e., ordinary disloca-

tions of all orientations, superdislocations, and twinning dislocations, bow out

between spatially distributed pinning points and move in a jerky way. This

can best be explained by a precipitation hardening mechanism controlling the

dislocation motion between room temperature and about 430

◦

C. At higher

temperatures, the dominating ordinary dislocations are smoothly bent and

move either in an unstable or in a viscous way. This may be interpreted by

the diﬀusion-controlled formation of Snoek atmospheres from antisite defects

and vacancies, which cause an additional friction by strain ageing eﬀects, thus

being responsible for the ﬂow stress anomaly. The occurrence of the anomaly

is connected with an inverse dependence of the strain rate sensitivity on the

strain rate or stress.

9.4 NiAl

The intermetallic compound NiAl has the B2 crystal structure. It shows a

strong plastic anisotropy. As demonstrated in Fig. 9.15, the material does

not exhibit a ﬂow stress anomaly along a soft deformation axis. Along the

hard axis, however, the ﬂow stress is several times higher than in the soft

orientations. Besides, it shows a weak anomaly, i.e., the ﬂow stress decreases

only slightly between room temperature and about 500

◦

C, as reviewed by

Miracle [562]. For applications at high temperatures, NiAl is alloyed with

metals like Nb and Ta to form two-phase alloys with strengthening precipitate

particles and grain boundary layers containing ternary phases, e.g., Laves

phases, as described by Sauthoﬀ [563,564].

9.4.1 Crystal Structure and Slip Geometry

Figure 9.18 presents the unit cell of an AB compound with the cubic B2 struc-

ture. One atom species A occupies the corners of the cube and the other atom

B the body-centered positions. The possible Burgers vectors are 100, 011,

and 111. Dislocations with the latter Burgers vectors are superdislocations.

The critical stress to move the dislocations is much lower for dislocations

9.4 NiAl 341

[001]

[010]

[100]

[111]

[011]

Fig. 9.18. Unit cell of the B2 structure with [100], [011], and [111] Burgers vectors

of the (0

11) slip plane

with 100 Burgers vectors than for the other two. Therefore, middle-oriented

specimens with reasonable orientation factors for 100 dislocations have a low

ﬂow stress. These orientations are called soft orientations. For the deformation

along the hard 100 directions, the 100 dislocations have zero orientation

factors so that the two other types of dislocations are activated. The ﬂow

stress is then much higher.

9.4.2 Microscopic Observations

The following experimental results are from in situ straining experiments

either on stoichiometric NiAl single crystals with (110) foil surfaces strained

along the soft [1

11] or [2

21] tensile directions [287], or on coarse-grained sto-

ichiometric polycrystals containing 0.2at% Ta [565]. The single crystals are

deformed exclusively by dislocations with the short 100 Burgers vectors.

The same holds for the NiAl-0.2at%Ta polycrystals at room temperature. At

higher temperatures, in NiAl-0.2at%Ta the dislocations with 110 Burgers

vectors were activated in the special orientation of the large grains during the

in situ straining experiments. Video 9.10 originates from the in situ fracture

study in [566].

At room temperature, the 100 dislocations with a large screw compo-

nent in both materials are strongly pinned as shown in Fig. 9.19. The average

distance between the pinning agents amounts to l =76nminNiAlandto

150 nm in NiAl-0.2at%Ta, corresponding to about 260 b and 500 b, respec-

tively. Between the pinning centers, the bowed-out segments are of angular

shape. The main features of this shape can be explained in terms of the line

tension model in anisotropic elasticity, which was described in Sect. 3.2.7.

Figure 9.20 shows the equilibrium shape of loops on a {110} plane, calculated

by (3.41) with elastic constants from [567] for determining the dislocation

energy factor K(β) and its derivative in the framework of anisotropic elasticity.

As pointed out earlier in [568], screw dislocations with b = 100 are unstable

on {100} planes. In screw orientation, the line energy E has a maximum in

contrast to elastic isotropy where it shows a minimum. At the same time, the

line tension Γ is negative, resulting in the instability. The ﬁgure shows that the