Messerschmidt U. Dislocation Dynamics During Plastic Deformation

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252 7 Ceramic Single Crystals

f(φ)=



sin

−

cos

π − 5φ



. (7.12)

To obtain the actual stress intensity of the crack of Fig. 7.22, the crack

opening proﬁle v(r) was measured and compared with the theoretical curve

v(r)=

1 − ν

√

2cr =

1 − ν



yielding K =0.55 MPa m

1/2

. This allows one to calculate proﬁles of equal

stress σ

around the crack. The proﬁle plotted in Fig. 7.22 is based on the ﬂow

stress of σ

= τ

∗

=2τ

∗

with τ

∗

= 115 MPa from Table 5.2, again with the

theoretical value of C = −1.61, which overestimates the true ﬂow stress but

may consider the irradiation hardening during the in situ experiment. In this

approximation, the curve ﬁts the width of the actual plastic zone but not its

elongated shape. The latter may depend on the slip geometry. The present slip

bands grow in length by glide of the dislocations generated by multiplication

of existing dislocations via the double-cross slip mechanism (Sects. 5.1.1 and

5.1.2). To increase the width of the bands would require extensive cross slip.

In conclusion, one can say, inside the plastic zone of the crack the high

stresses predicted by the elastic theory of fracture are relaxed down to the

average ﬂow stress of the material owing to dislocation multiplication and

motion. The shape of the plastic zone depends on the ability of the dislocations

to slip and cross slip.

In the literature, the question had been discussed whether in crystals of

low dislocation density, cracks may grow accompanied with a dislocation-free

zone or whether dislocations are nucleated in front of the crack due to the high

stress concentration. Respective in situ straining experiments in a TEM were

performed and interpreted by Ohr [422]. In these experiments, dislocations are

generated mainly under shear loading, in contrast to the experiment described

above with tensile loading.

7.2.7 Summary

MgO single crystals with small precipitates have proved an ideal material

to experimentally study the statistical and kinematic aspects of the interac-

tion between dislocations and localized obstacles that obey Friedel statistics.

There are no other experimental data found in the literature on the frequency

distribution of the forces acting on the obstacles, the relation between the

curvature of bowed-out dislocation segments, and their length as well as the

Friedel relation between the obstacle distance and the eﬀective stress. The

material shows all the three ranges of mechanisms presented schematically in

Fig. 5.25, the Peierls mechanism at low temperatures, the thermally activated

overcoming of obstacles at and above room temperature, and the dominance

of the athermal stress component owing to long-range dislocation interactions

at high temperatures.

7.3 Zirconia Single Crystals 253

7.3 Zirconia Single Crystals

Zirconia may serve as a constituent of materials with attractive properties

for structural applications over a wide range of temperatures. The tetragonal-

to-monoclinic transformation is used for designing tough materials around

ambient temperature [423, 424]. At high temperatures, partially stabilized

zirconia (PSZ) may have a very high yield stress due to precipitation hard-

ening (e.g., [425, 426]). In addition, ferroelastic deformation observed in

so-called t



zirconia may contribute to a high toughness at high temperatures

(e.g., [427, 428]). This chapter is concerned with the dislocation processes in

cubic zirconia single crystals, the ferroelastic deformation in t



and partially

stabilized zirconia, and the subsequent plastic deformation.

7.3.1 Crystal Structure and Slip Geometry of ZrO

–Y

alloys

Heuer and coworkers have presented an early review of the structure and

microstructure of ZrO

–Y

materials [429]. Pure zirconia (ZrO

)hasthe

CaF

structure at temperatures above about 2,400

◦

C as shown in Fig. 7.24.

While the zirconium ions occupy the sites in a face-centered cubic unit cell, the

oxygen ions are situated in 1/4111 and equivalent positions. Below 2,400

◦

ZrO

undergoes a phase transformation to the tetragonal phase. In addition

to a small elongation by about 2% in the direction of the c axis, rows of oxygen

ions are shifted in upward and downward directions as indicated by the arrows

in the ﬁgure.

These high-temperature phases can be stabilized down to low temperatures

by adding aliovalent cations. According to the ZrO

–Y

phase diagram,

the cubic phase (c-ZrO

) is stabilized at Y

concentrations higher than

about 8–10 mol% (fully stabilized zirconia, FSZ). At lower yttria concentra-

tions, yttria-lean precipitates of the tetragonal structure form in an yttria-rich

zirconium

oxygen

a,c

Fig. 7.24. Crystal structure of ZrO

single crystals. Cubic ZrO

: oxygen ions in the

drawn 1/4111 positions. Tetragonal ZrO

: rows of oxygen ions shifted as indicated

by arrows

254 7 Ceramic Single Crystals

cubic matrix (partially stabilized zirconia, PSZ). The diﬀusion-controlled pre-

cipitation reaction proceeds very slowly owing to the sluggish cation diﬀusion

requiring long times to attain the equilibrium concentration of the precip-

itated tetragonal phase. A variety of microstructures may form depending

on the yttria concentration, annealing temperature, and duration [430]. The

diﬀusive transformation cannot take place during quenching from the cubic

phase ﬁeld. The crystals then undergo a displacive phase transformation to a

metastable tetragonal material of uniform yttria concentration, the so-called

tetragonal polydomain zirconia or t



zirconia.

As cubic zirconia may constitute the matrix of the two-phase materials,

its deformation behavior forms the basis of the understanding also of the

deformation of the more complex materials. The Burgers vectors in c-ZrO

are of type 1/2110 with slip on the easy {100} planes. Most experiments

are carried out along a soft 211 loading axis favoring single slip on one

cube system with an orientation factor of m

=0.47. At high temperatures,

{110} and {111} planes may be activated as secondary planes with orientation

factors of 0.5 and 0.41. These planes are also activated if cube slip is suppressed

by a hard 100 deformation axis.

7.3.2 Microscopic Observations in Cubic ZrO

Zirconia single crystals are brittle at low temperatures. The lowest temper-

atures where plastic deformation of cubic zirconia with 10 mol% yttria in

compression in the soft orientation was successful is 400

◦

C under usual defor-

mation conditions [213], and 250

◦

C under conﬁning hydrostatic pressure [214].

All ﬁgures and data following refer to this material if not stated otherwise. As

shown in Fig. 7.25 for 500

◦

C, slip is localized in narrow slip bands consisting

mainly of screw dislocations pinned at localized obstacles. Most obstacles are

supposed to be small precipitates, probably containing nitrogen [431]. The

obstacle distances l are of the order of magnitude of 0.1 μm and decrease with

decreasing temperature and thus with increasing eﬀective stress, but they do

not fulﬁl the Friedel relation (4.54), probably because of a diﬀerent obstacle

spectrum at diﬀerent temperatures and because of the presence of many jogs

marked J, trailing dislocation debris. Similar dislocation structures are found

also at 700

◦

C as presented in Fig. 7.26, where the TEM foil was cut parallel

to the primary slip plane.

At higher temperatures, the localized obstacles are no longer active . The

dislocation structure is quite homogeneous. The dislocations being pinned by

jogs in their screw components bow out to larger arcs as demonstrated in

Fig. 7.27 for 1,000

◦

C. Stereo pairs reveal that the dislocations are arranged

not solely on cube planes and that the cross slip planes probably are {111}

planes [432]. In addition, many dislocation loops are present.

In the same temperature range, also in situ straining experiments in an

HVEM were successful [282]. As shown in Fig. 7.28, the dislocation structures

consist of long edge dislocations and screw dislocations pinned by jogs in the

7.3 Zirconia Single Crystals 255

0.5 µm

Fig. 7.25. Screw dislocations pinned at localized obstacles in c-ZrO

deformed along

[

2] at 500

◦

C. (

111) foil plane. Jogs trailing debris dipoles are marked J.Fromthe

work in [213]

1 µm

Fig. 7.26. Screw dislocations pinned at localized obstacles in c-ZrO

deformed along

[

2] at 700

◦

C. Foil plane parallel to (001) slip plane. Jogs trailing debris are marked

by arrows. From the work in [213]

“α” like conﬁguration typical of dislocation multiplication by the double-cross

slip mechanism (Fig. 5.5). The equilibrium shape of the dislocations can be cal-

culated by the line tension approximation in anisotropic crystals as described

256 7 Ceramic Single Crystals

1 µm

Fig. 7.27. Dislocation structure in c-ZrO

after macroscopic deformation along

[

2] at 1, 000

◦

C. Specimen cooled under load. b projection of Burgers vector. From

the work in [433]

(a)

(b)

0.5 μm

Fig. 7.28. Dislocation structure during in situ straining of c-ZrO

deformed along

[

2] at 1, 150

◦

C. (

111) foil plane. Long edge dislocations (a) and screw dislocations

with an “α”-like conﬁguration (b). Inset in (b): Equilibrium conﬁguration of dislo-

cation loop on the {100} slip plane in the projection of the micrographs. From the

work in [282]

in Sect. 3.2.7. The inset in Fig. 7.28b presents the slightly angular shape of the

loops on the {100} slip plane with relatively straight segments near the cube

orientations. The curved dislocations approximate the calculated shape, which

is also obvious from Fig. 7.27. Fitting the size of the loops to the experimental

images and using (5.18), the back stress τ

caused by the dislocation curvature

can be calculated. For the constant C, the theoretical value of −1.61 may be

7.3 Zirconia Single Crystals 257

more suitable than the experimental value of −5.19 (see Sect. 3.2.7) because of

the large size of the loops compared to small bow-outs at dislocations pinned

by localized obstacles. This leads to an upper limit of τ

=75MPa.

The video recordings reveal the dynamic character of the dislocation

motion on the easy cube plane at 1,150

◦

Video 7.3. Dislocation motion in c-ZrO

on a {001} plane during deformation

along [

2] at 1, 150

◦

C: This video consists of four short sequences. The lines b

and t at the beginning mark the directions of the projection of the [1

10] Burgers

vector and the [110] trace of the (001) slip plane on the surface. The imaging is

frequently obscured by strong fringes caused by the bending in the slip bands. In

specimens with a [

2] tensile axis and a (

111) foil surface, the easy primary slip

system with a (001) cube slip plane is activated. Dislocations on this system move in

a very jerky way frequently over distances larger than the frame width so that the

dislocations appear or disappear within two successive frames. Figure 7.29 shows

the disappearance of the dislocation marked by the arrow between (a) and (b) and

the sudden formation of a small loop between (c) and (d). In the second sequence,

a slip band proceeds forward always followed by the bending fringe.

Thus, dislocation motion on the cube plane at 1,150

◦

C is obviously controlled

by the long-range internal stress ﬁelds. It is of athermal nature.

At high temperatures, the dislocation structures become very homoge-

neous, and consist of a network of dislocations on diﬀerent slip systems, as

demonstrated by Fig. 7.30. This points at the action of recovery-controlled

deformation mechanisms.

(a) (b)

Fig. 7.29. Disappearance of a dislocation marked by an arrow within successive

frames (a)and(b) of the video recording, and sudden appearance of a dislocation

loop between (c)and(d)

258 7 Ceramic Single Crystals

1 µm

Fig. 7.30. Dislocation structure in c-ZrO

after macroscopic deformation along [

at 1,400

◦

C. Specimen cooled under load. CD projection of compression direction.

From the work in [434]

In the hard [100] orientation in which the slip systems with {100} planes

have zero orientation factors, dislocations on {111} as well as on {110} planes

are activated. Figure 7.31 presents an example with two sets of dislocations

on {111} planes. In Fig. 7.31a, both sets are visible, while in Fig. 7.31b the

horizontal set is extinguished. At the relatively low temperature of 800

◦

C, the

dislocations are pinned by localized obstacles and bow out between them as on

{100} planes during the deformation along the [

2] loading axis. Some debris

marked D is also produced. On the non-cube planes, these structures prevail

up to higher temperatures. Above 1,200

◦

C, three-dimensional networks form,

which are characteristic of the high-temperature deformation.

Non-cube slip was also studied by in situ straining experiments in the

HVEM at 1,150

◦

C. An unusual (211) slip plane was observed in specimens

deformed along the soft [

2] orientation but with a (110) foil surface, as

demonstrated by the following video clip.

Video 7.4. Dislocation processes in c-ZrO

on a (211) plane during in situ strain-

ing along [

2] at 1, 150

◦

C: The sequence presents dislocation motion in a specimen

with a (110) foil surface. The Burgers vector is now 1/2[0

11] and the trace t is that of

the (211) slip plane. The dislocations are smoothly curved at this temperature. Fre-

quently, they move in a jerky way over distances comparable with those between the

dislocations. In contrast to the motion on the {001} easy slip plane, the dislocations

also move slowly in a viscous manner. Thus, the movement is controlled by both the

7.3 Zirconia Single Crystals 259

Fig. 7.31. Dislocation structure of c-ZrO

deformed along d = [100] at 800

◦

Cup

to 1.1% imaged under diﬀerent g vectors at the [001] pole. g = (200) (a), g =(2

20)

(b). From the work in [435]

long-range internal stress ﬁelds and a viscous friction process. At A, two half-loops

meet and partially annihilate each other. As the slip planes are not identical, jogs

remain in both dislocations, which impede further motion so that the dislocations

bow out. Simultaneously, a jog in a dislocation at B trails a dipole (debris) in low

contrast, which later on opens to a wider dislocation loop. The dislocation at C

moving to the right changes its slip plane a few times, visible by steps in the lower

trace.

One in situ experiment was successful with the hard [100] tensile direction.

Video 7.5. Dislocation motion in c-ZrO

on a {10

1} plane during deformation

along [100] at 1, 150

◦

C: Loading along [100] activates dislocations with 1/2[1

10]

Burgers vectors to move on {110} planes in a viscous way, similar to the motion

on {211} planes. At A, a dislocation multiplies and emits two new dislocations in

260 7 Ceramic Single Crystals

opposite directions. A dislocation loop is formed at B. Unfortunately, strong contrast

changes occur due to corrections of the imaging conditions.

To estimate the athermal stress component, dislocation densities were mea-

sured from TEM images taken at several temperatures. In specimens deformed

in the soft orientation, the densities amount to about 2 × 10

−2

over a

wide temperature range, but they rise drastically below 700

◦

C. In the hard

orientation, the densities equal about 10

−2

. They strongly decrease above

1,200

◦

C. The calculation of the internal stress will be described later.

In conclusion, one may say that dislocation motion is impeded by small

precipitates acting as localized obstacles at and above the lowest temperatures

where TEM preparation was successful, that is, 500

◦

C for slip on the soft cube

plane and 700

◦

C for slip on the {111} and {110}planes. At and above 1,000

◦

dislocation segments on the cube planes are long, smoothly curved, and pinned

by jogs. They move in a very jerky athermal way. At 1,150

◦

C, dislocations

on non-cube planes are also smoothly bent, but they move in a continuous

viscous way. At high temperatures, that means, at 1,400

◦

C, the dislocations

form three-dimensional networks in all orientations, which suggests that the

deformation is controlled by recovery processes.

7.3.3 Dislocation Dynamics in Cubic ZrO

The dependence of dislocation velocities on the stress was measured between

1,100

◦

C and 1,450

◦

C by the stress pulse etching technique on dislocations

introduced by microhardness indents [21]. While edge dislocations are slightly

more mobile at lower temperatures, screw dislocations are faster at higher

ones. This is in accordance with the fact that screw dislocations predominate

in specimens deformed at low temperatures, whereas edge dislocations prevail

at high ones. The stress exponent m



can be obtained from double-logarithmic

plots of the dislocation velocity vs. stress, as shown in Fig. 7.32. At 1,435

◦

Fig. 7.32. Stress dependence of edge dislocation velocities at diﬀerent temperatures

in ZrO

–10 mol% Y

, measured by the stress pulse etching technique. Data from

[21]

7.3 Zirconia Single Crystals 261



is about 3–5, but at 1,100

◦

C, m



=1.4. The apparent activation energies

calculated from Arrhenius plots of dislocation velocities against the reciprocal

temperature are of the order of magnitude of 5 eV. The stress range of these

measurements touches that of the macroscopic experiments described below

only at the high-temperature end. A comparison with the macroscopic data

reveals that at high temperatures both the stress exponent and the activation

energies are of the same order of magnitude as the data for creep between

1,400 and 1,500

◦

C (e.g., [436]). As it will be described in the following section,

these data are interpreted by diﬀusion-controlled recovery, but not by the

dislocation mobility. The low-temperature end of the dislocation velocity data

corresponds to the intermediate temperature range discussed below. However,

the stress range of the velocity measurements is far below the macroscopic ﬂow

stress. Thus, these data are not suited to interpret the macroscopic plasticity

data.

7.3.4 Macroscopic Deformation Properties of Cubic ZrO

Originally, cubic zirconia single crystals were deformed between 1,200 and

1,500

◦

C (e.g., [437,438]). They were considered brittle below this range. How-

ever, careful specimen preparation and a low strain rate of 10

−6

−1

enabled

the deformation of these crystals in the soft orientation down to 400

◦

[213, 324] and in the hard orientation down to 500

◦

C [435]. Under conﬁn-

ing hydrostatic pressure, the material can be deformed even down to 250

◦

[214]. In addition to the crystals with 10 mol %Y

, some experiments were

carried out on materials with 15 and 20 mol %Y

The stress–strain curves are mostly smooth except for the materials with

higher yttria concentrations where plastic instabilities occur at intermedi-

ate temperatures. Some features of the instabilities were discussed before in

Sect. 5.3. Particularly at low temperatures, the curves show a strong yield drop

eﬀect. The strain hardening coeﬃcient is always low. During almost steady

state deformation, the stress–strain curves were interrupted by stress relax-

ation, strain rate cycling, and temperature change tests. The yield stress σ

is plotted in Fig. 7.33 as a function of temperature. The yield stresses along

112 are always lower than those along 100 except at the highest tempera-

ture of 1,400

◦

C. The 15 and 20 mol% materials, too, have higher yield stresses.

The curves show several linear ranges with diﬀerent temperature sensitivities

Δσ/ΔT . In general, Δσ/ΔT is higher for lower temperatures and for the hard

100 orientation.

Strain rate sensitivities were mostly determined by stress relaxation (SR)

tests. The results may be plotted as the logarithm of the relaxation rate vs.

the stress. As the relaxation rate is proportional to the plastic strain rate,

the reciprocal slope of the curves represents the strain rate sensitivity r,as

described above in Sect. 2.1. Normal stress relaxation curves originating from

the thermally activated overcoming of obstacles are bent towards the stress

axis as plotted in Fig. 7.34a, indicating a decreasing strain rate sensitivity