Messerschmidt U. Dislocation Dynamics During Plastic Deformation

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352 9 Intermetallic Alloys

Table 9.3. Decomposition of dislocations with [110] Burgers vectors on the (1

10)

plane in NiAl

Reaction (9.3) (9.4)

Total disl.

Line orient., φ [00

1], 90

◦

[111], 35.3

◦

(10

−9

N) 1.201 0.945

Comp. disls.

BV [100] [010] [100] [010]

Plane, φ (010), 90

◦

(100), 90

◦

11), 54.7

◦

(10

1), 54.7

◦

(10

−9

N) 0.666 0.666 0.694 0.694

Energy gain

−



(10

−9

N) −0.13 −0.44

Fig. 9.27. Motion of a macrokink in a dislocation with 110 Burgers vector

decomposed according to (9.3). After [583]

The geometry and energy balance of both reactions are summarized in

Table 9.3. The energies are the prelogarithmic factors E

calculated in the

framework of anisotropic elasticity. According to the table, the reaction (9.4)

along 111 is slightly unfavorable in terms of the elastic energy factors of the

dislocations involved. The interaction energy is not considered.

Mills et al. [583] present a model of the diﬀusional motion of an edge

dislocation decomposed according to (9.3) by the spreading of constricted

macrokinks. They assume that the dislocation forms a constricted macrokink

of height h as outlined in Fig. 9.27. b

is the 110 total Burgers vector, and

and b

are the 100 Burgers vectors of the components. To shift the

macrokink by a distance ΔL, the area A has to be swept by the dislocation

with b

acting as a vacancy source while the area B is swept by absorbing

these vacancies. Under very simplifying assumptions, the change in energy by

shifting the macrokink by a distance dL is

dW =(τb

h + σb

d + E

110

− 2E

100

)dL. (9.5)

Here, τ is the shear stress acting on the macrokink with 110 Burgers vector.

This kink is of screw character. σ is the normal stress along a 100 direction

causing the dislocation with b

to climb, whereas the dislocation with b

does

not experience a climb force from the external stress. Finally, d is the width

over which the component dislocations have decomposed. The ﬁrst term in

9.4 NiAl 353

(9.5) is the gain in energy by shifting the macrokink, the second one is the

loss in energy to form the vacancies on the dislocation with b

. As described

above, the change in the self-energy by the decomposition E

110

−2E

100

is zero

for the decomposed edge dislocation.

During the motion of the macrokink by dL, a number of vacancies dN

have to be transferred from the vacancy source at the border of the area A

of the dislocation with Burgers vector b

to the vacancy sink at B of the

dislocation with b

, which equals

dN =

dL,

where a is the lattice parameter. The change in the elastic energy per

transferred vacancy is

w = −

=(τb

h − σb

Applying (4.83) with Δ (ΔG

)=w, the vacancy supersaturation at the source

dislocation is

Δc

= c

− c

= c



exp





− 1



with c

being the thermal equilibrium concentration of vacancies. In the limit

of Δc

 c

,thereis

Δc

≈





In a simple model, the vacancy ﬂux from the source at A having a vacancy

concentration c

−Δc

to the sink at B with a concentration c

+Δc

occurs

by pipe diﬀusion along the component dislocations. Applying Fick’s law (4.91)

and considering that τ = σ/2 in the hard orientation, ﬁnally yields



√

2 − d



(9.6)

for the velocity of the macrokink. D

is the pipe diﬀusion coeﬃcient. The

macrokink velocity exhibits a strong temperature dependence via the pipe

diﬀusion coeﬃcient. It depends on the geometric parameters of the kink and

decomposition width, where the latter is particularly important. This param-

eter will further be considered in Sect. 9.5. Mills et al. [583] observed a good

agreement with experimental data by applying the geometric parameters of

the microstructural investigations and macroscopic data at a temperature of

600

◦

C, and by assuming that the activation energy of pipe diﬀusion is half

of that of bulk diﬀusion. For the mixed dislocations oriented along 111,the

motion occurs by combined glide and conservative climb. The character of the

climb-controlled motion is not jerky, but viscous.

In a later model [571], it was considered that most dislocations orient

along 111 and that macrokinks have not been observed. According to the

354 9 Intermetallic Alloys

revised model, the movement of the dislocation consists of a succession of climb

and glide of the 100 component dislocations, as sketched in Fig. 9.28. The

forces acting on the product dislocations are the components of the external

stress, the elastic interaction between the products, a chemical force due to

the point defect super- or subsaturation and, in case of bowed segments, the

line tension. In the hard orientation, the external tensile or compressive stress

causes a pure climb force on one component dislocation and no force on the

other. This would lead to an increase in the decomposition width. The elastic

interaction between the components tries to arrange them perpendicular to

the glide plane of the total dislocation and then cause a radial attraction. In

addition, the dislocation climbing under the external stress changes the point

defect concentration away from the equilibrium, which results in a chemical

force on the other product dislocation. If the point defect ﬂux generated at

one component is totally annihilated at the other one (conservative climb),

the dislocation couple will move along the glide plane of the nondecomposed

parent dislocation. The deviation of the dislocation along 111 from the edge

orientation results in a superposition of glide of the components on their low-

index glide planes. These processes may occur on an atomic scale as assumed

in the scheme of Fig. 9.28.

Srinivasan et al. [571] estimate the dislocation velocity by assuming the

climb to be diﬀusion-controlled (see Sect. 4.10.4). They arrive at a formula very

similar to (4.92). Instead of the cut-oﬀ radius R in the logarithmic term there

appears the decomposition width d. Accordingly, the velocity depends strongly

on the temperature via the diﬀusion coeﬃcient and weakly on the decompo-

sition width. Inserting data for the deformation at about 600

◦

C, the (bulk)

diﬀusion coeﬃcient of Ni at that temperature, an experimental decomposition

width of 2 nm for the dislocations aligned along 111, and a characteristic dis-

location density of 2 × 10

−2

yields a dislocation velocity in the 10

−5

−1

range, only one order of magnitude lower than that of the ﬂow stress data. This

glide

climb

1/2

Fig. 9.28. Schematic drawing of the combined climb and glide motion of a decom-

posed dislocation with [110] Burgers vector arranged along [111] in NiAl. The image

plane is the (111) plane perpendicular to the extension of the dislocation. P is the

direction of propagation, d the decomposition width. The edge dislocation symbols

indicate the projections of the edge components of the 100 product dislocations. σ

is the projection of the applied stress onto the image plane. The open square marks

a diﬀusing vacancy. After [571]

9.5 FeAl 355

estimation is certainly no proof of the model. In particular, the assumption

of diﬀusion-controlled climb is questionable since it requires the characteristic

diﬀusion length to be great compared to the jog distance. This is certainly

not fulﬁlled at the small separation of the component dislocations. Besides,

the coordinated diﬀusion during the conservative climb in the elastic stress

ﬁeld of the composite dislocation may be controlled rather by pipe diﬀusion

than by bulk diﬀusion. Thus, a comprehensive theory of the motion of these

dislocations still has to be established.

9.4.5 Summary

The dislocations with 100 Burgers vectors in NiAl, which are activated by

the deformation near room temperature along soft orientations, are pinned by

high jogs in the screw components. The jogs are created by double-cross slip.

They are stabilized against lateral motion and annihilation by lattice friction.

This is exactly the model suggested by Sriram et al. [549] for TiAl. It does

not lead to a ﬂow stress anomaly. At elevated temperatures, the 100 disloca-

tions move in a viscous way pointing at the operation of a diﬀusion-controlled

mechanism. In the hard orientation, dislocations with 111 and 110 Burg-

ers vectors are active at high temperatures. When moving on {110} planes,

the 110 dislocations may be very straight and oriented along the edge and

mixed 111 directions. As suggested by Srinivasan et al. [571] and in the

present book, the straight oriented arrangement may result from a decompo-

sition into dislocations with 100 Burgers vectors. The preferred orientations

are the intersection lines between the glide planes of the parent dislocation

and low-index glide planes of the product dislocations. The decomposition may

occur by glide and is therefore diﬀerent from the climb dissociation observed

by H. Saka et al. in several materials (e.g., in β brass [584]). The decom-

posed dislocations cannot move by glide alone but by the aid of conservative

climb, i.e., a diﬀusional ﬂow of vacancies from one product dislocation to the

other. It will be discussed in the following sections that this model need not

be restricted to dislocations with 110 Burgers vectors in NiAl, but may be

more general.

9.5 FeAl

Like NiAl, FeAl has the B2 structure with 111 and 100 Burgers vectors.

The dominating slip planes are of {110} type. FeAl single crystals show a ﬂow

stress anomaly with a maximum at about 550

◦

C as shown in Fig. 4.40.

9.5.1 Microscopic Observations

The following data originate from experiments on an Fe-43at%Al single crystal

[8], which was annealed at 400

◦

C for 1 week followed by slow cooling to room

356 9 Intermetallic Alloys

temperature to anneal out the excess point defects originating from crystal

growth. Specimens for in situ straining experiments were cut with a [753]

tensile axis and (1

21) foil planes. Samples for macroscopic compression tests

were prepared with the same loading axis and a (13

50 53) side face about

◦

oﬀ (0

11). This orientation favors slip on the (101)[11

1] system. Notable

orientation factors also have (011)[11

1], (101)[010], and (011)[100].

In FeAl, there is a transition of the dominating Burgers vector from 111

to 100 [585,586] just at the temperature of the ﬂow stress peak, which might

cause the anomaly [587]. However, rapid quenching after deformation showed

that dislocations with 111 Burgers vectors dominate during deformation at

the high strain rate of 5 × 10

−2

−1

even at the high temperature of about

650

◦

C [588]. Similarly, it was observed that the deformation at the peak tem-

perature of 550

◦

C started by dislocations with 111 Burgers vectors, whereas

those with 100 Burgers vectors appeared only after a few percent of plastic

strain [589]. Conventional TEM in the high-voltage electron microscope of

macroscopically deformed specimens was performed to secure that the defor-

mation of the present material is carried by dislocations of the same Burgers

vector up to the ﬂow stress maximum. In Fig. 4.40, the temperatures at which

dislocation structures were studied are marked by dashed grey vertical lines.

The microstructure with dislocations of 111 Burgers vectors after defor-

mation at 495

◦

C was presented in Fig. 3.22. Dislocations with these Burgers

vectors prevail up to the ﬂow stress maximum. Stereo image pairs show that

most dislocations are not conﬁned to their slip planes indicating the impor-

tance of climb. Further deformation at 564

◦

C activates dislocations with 100

Burgers vectors as demonstrated before in Fig. 1.4.

Due to elastic instabilities, many dislocation lines of both Burgers vectors

exhibit an angular shape with a sharp knee. Some of them are indicated in the

ﬁgures. If the overall orientation of a dislocation lies in the unstable range, the

dislocation decomposes into stable segments as labeled by A in Fig. 3.22. In

[590], a comprehensive study was given of the elastic properties of dislocations

with all relevant Burgers vectors in B2 Fe–Al of diﬀerent composition. Accord-

ing to that, dislocations with 111 and 100 Burgers vectors are elastically

unstable in certain orientation ranges. The insets of Figs. 3.22 and 1.4 present

the calculated shapes of dislocation loops on the (101) plane using the line

tension model in anisotropic elasticity with elastic constants from [591]. The

line tension maxima and minima of the 111 dislocations are not located at

the screw and edge orientations. They diﬀer by a factor of about 8, resulting in

the elongated shape of the equilibrium loop. Between about 38

◦

and 55

◦

, i.e.,

for orientations close to [10

1], the dislocations are unstable resulting in edges

in the dislocation line. These knees and preferential orientations are evident

in Fig. 3.22, which also shows many mixed dislocations having decomposed

into a zig-zag shape, e.g., at A. The energy of the [010] dislocation shows an

anomalous decrease with increasing edge character with its minimum at about

125

◦

. The shape of this dislocation is symmetric around the Burgers vector

direction with relatively ﬂat segments at the two 111 orientations. The screw

9.5 FeAl 357

0.5 µm

Fig. 9.29. Dislocations in Fe-43Al moving during an in situ straining experiment

at room temperature. g =(10

1) near the [1

21] pole. g is parallel to the trace of the

(101) primary slip plane. b

and b

are the projections of the [11

1] and [

111] Burgers

vectors. From [8]. Copyright Elsevier Ltd. (2005)

orientation is unstable in the range between about 14

◦

and −14

◦

.Theinsta-

bility of the screw direction and the preferential orientations are obvious from

Fig. 1.4. Apparently, the edge orientation seems to be another preferential

orientation which is not described by the elastic dislocation properties.

In Fig. 4.40, the temperatures at which in situ straining experiments

were performed in the HVEM are indicated by black vertical lines. At

room temperature, dislocation motion was observed in the plastic zone of

a crack. Figure 9.29 shows the dislocation structure during deformation. The

projections of the Burgers vectors are marked. Moving near-screw dislocations

with Burgers vector b

are labeled by arrows. In the upper part of the ﬁgure,

long dislocations of 30

◦

character have developed. All dislocations bow out

between cusps in the dislocation lines. The distances l between the obstacles

are in the range of 0.1 μm.

The dynamic behavior of the dislocations is demonstrated in the following

video sequences.

Video 9.12. Motion of dislocations with 111 Burgers vectors in Fe-43at%Al at

room temperature: The clip consists of four sequences. Compared to the others, the

ﬁrst one is taken at a lower magniﬁcation. The dislocations move in a jerky way

over distances longer than the segment length between the pinning points. The long

dislocations are of near-screw character.

In situ experiments at 363

◦

C, i.e., when the ﬂow stress begins to increase,

show straight slip traces belonging to glide on {110} planes, with cross slip

events between diﬀerent such planes. The dislocations with 111 Burgers vec-

tors consist of straight segments of two orientations. One is consistent with

358 9 Intermetallic Alloys

1 µm

Fig. 9.30. Slip traces and dislocations with 111 Burgers vectors moving during

in situ straining in Fe-43at%Al at 530

◦

C. g =(10

1) near the [1

21] pole. From [8].

the screw direction, whereas the other one corresponds to a mixed orientation

with a large edge component, which may be 111 on the respective {101}

plane. At 530

◦

C, i.e., in the anomaly range of macroscopic deformation tests,

the slip traces (thin arrows in Fig. 9.30) are curved pointing at the superpo-

sition of climb or cross slip in addition to glide. Nevertheless, the traces are

close to those of the (011) planes. The dislocations (thick arrows) imaged with

the (10

1) reﬂection belong to the primary [11

1] Burgers vectors. The projec-

tions of the straight segments are again consistent with the screw orientation

and the 60

◦

character on the (011) plane. The next video shows the motion

of such dislocations.

Video 9.13. Motion of dislocations with 111 Burgers vectors in Fe-43at%Al at

530

◦

C: The dislocations of angular shape move very slowly in a viscous way.

In addition, slip bands occur with dislocations that are extinguished at

g =(10

1). These dislocations can be imaged with a (211) g vector at the

31] pole as demonstrated in Fig. 9.31 taken at 447

◦

C. They are also invisible

with g = (200) so that their Burgers vector is [010]. The orientation of the slip

traces suggests again (101) slip planes. These dislocations are straight, with

their projections corresponding to 111 directions, i.e., the angle between the

Burgers vectors and the dislocation lines is 54.7

◦

. Their motion is recorded in

9.5 FeAl 359

1µm

Fig. 9.31. Dislocations with 100 Burgers vectors moving during in situ deforma-

tion of Fe-43Al at 447

◦

C. g = (112) near the [1

31] pole. Projections p

=[71

4] of

[

111] direction and p

417] of [11

1] direction. From [8]. Copyright Elsevier Ltd.

(2005)

Video 9.14. They move and multiply in bands separated from the bands with

111 dislocations.

Video 9.14. Motion of dislocations with 100 Burgers vectors in Fe-43at%Al at

447 and 530

◦

C: The dislocations of a straight oriented shape move viscously. Even

at a temperature lower than in Video 9.13, the 100 dislocations move faster than

those with 111 Burgers vectors. After unloading, they relax into a curved shape.

At high temperatures far above the ﬂow stress maximum (near 700

◦

C),

the dislocations show a totally diﬀerent behavior, which is documented in the

following video.

Video 9.15. Dynamic formation and annihilation of dislocations in Fe-43at%Al

at 700

◦

C: Smoothly curved dislocations move viscously in a very irregular way

pointing at a combination of glide with rapid climb. New dislocation length is cre-

ated continuously by certain segments bowing out. On the other hand, dislocation

segments coming across each other frequently annihilate, so that the dislocation den-

sity remains constant on a low level over a long time of observation. At L near the

end of the sequence, a small loop is created by a moving dislocation, which shortly

afterward collapses again. The loop is also shown in Fig. 9.32. When the dislocations

stop moving, they form straight segments, the projections of which are consistent

with the three 100 line directions.

360 9 Intermetallic Alloys

1 µm

Fig. 9.32. Frame from Video 9.15 showing the blurred image of a moving dislocation

marked by small arrows and a dislocation loop indicated by a thick arrow.From[8].

The irregular nonplanar motion of the dislocations suggests that, at this

high temperature, climb is not only a process of recovery but also a mode of

dislocation motion.

9.5.2 Macroscopic Deformation Parameters

Figure 9.33 shows two stress–strain curves at 550

◦

C with sections taken at

the diﬀerent strain rates indicated. The curves are either smooth or serrated

depending on the strain rate and, in general, also on the temperature. Smooth

deformation curves occur only at room temperature for strain rates above

−6

−1

, and above about 550

◦

for strain rates less than 10

−3

−1

.Below

550

◦

C, optical microscopy shows single slip on localized bands corresponding

to the (101) primary slip plane. Above that temperature, traces are revealed

belonging to the (011) plane.

As mentioned earlier, the temperature dependence of the yield stress

was presented before in Fig. 4.40. The curve consists of three ranges, a low-

temperature range of a normal temperature dependence, i.e., with a slightly

decreasing ﬂow stress between room temperature and about 350

◦

C, a range of

an anomalous increase between about 350 and 550

◦

C, and a high-temperature

range above 550

◦

C with again a normal decrease of the ﬂow stress. In this

range, very strong yield points occur. Figure 4.40 presents the lower yield

9.5 FeAl 361

Fig. 9.33. Stress–strain curves of Fe-43at%Al deformed at 550

◦

C at diﬀerent strain

rates ˙ε. R are stress relaxation tests. Data from [8]

Fig. 9.34. Dependence of the load to achieve dislocation velocities suitable for obser-

vation in the HVEM for three in situ experiments including temperature changes.

Each symbol belongs to one experiment. Data from [8]

points. The upper yield point of 342 MPa at 594

◦

C has almost the same value

as the stress maximum at 550

◦

The tensile stage for the in situ straining experiments in the HVEM allowed

the measurement of the force acting on the micro-specimens. During the exper-

iments at higher temperatures, several times a single specimen was deformed

successively at diﬀerent temperatures. The straining stage was controlled in

such a way that dislocation velocities were achieved in the range that can well

be recorded (about 10–1,000nm s

−1

). In Fig. 9.34, these loads are plotted ver-

sus the temperature for three in situ experiments. Each symbol corresponds

to one specimen deformed at diﬀerent temperatures. In all experiments, there

is a continuous decrease of the loads with increasing temperature, i.e., the

anomalous increase of the ﬂow stress observed in macroscopic experiments

did not occur during the in situ tests.

The strain rate sensitivity of the ﬂow stress was studied by stress relaxation

tests. The relaxation curves are quite straight or have the usual curvature