Messerschmidt U. Dislocation Dynamics During Plastic Deformation

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272 7 Ceramic Single Crystals

with τ

being the stress at the beginning of the relaxation and

A =

mRτ

(m − n)(1+Θ

/S)˙γ

1/n)

S is the elastic stiﬀness deﬁned in Sect. 2.1. At the beginning of relaxation,

that is, in stage 1 of the relaxation curve,



=(dln˙γ/dlnτ)

τ =τ

= m (1 − A)+nA, (7.18)

which is close to m for large m (and small A). At the end of the relaxation,

that means in stage 2,



=(dln˙γ/dlnτ)

τ τ

= n. (7.19)

Thus, the two-stage relaxation curves plotted on a double-logarithmic scale

allow one to separate the dynamic dislocation properties from the recovery

behavior. Such curves occur also in other materials, for example, in MgO in

the high-temperature range. In the following, some features of the deformation

behavior described in Sect. 7.3.4 will be discussed qualitatively on the basis

of the model.

The stress exponents of the relaxation curves of deformation along 112 at

low strains and suﬃciently high starting strain rates are about 100 in stage 1

and 4 in stage 2. The high stress exponent in stage 1 approximates the dynamic

stress exponent m and describes athermal dislocation motion characteristic

also of lower temperatures where the strain rate sensitivity is very low. The

strain softening exponent of 4 in stage 2 agrees with the stress exponents

derived from incremental load changes performed during creep at small strain

rates so that the dynamic stress exponent m could not be observed (e.g.,

[446]). In the double-logarithmic plot, relaxation curves during deformation

along 100 show only a slightly inverse shape with a starting slope of m



≈ 9

and a ﬁnal slope of m



≈ 6. This prevents a clear separation of the processes

but ﬁts the viscous dislocation motion on non-cube planes.

For deformation along 112, the activation enthalpies at about 1,400

◦

amount to 5.5–6 eV. This is close to the values of cation self-diﬀusion in the

bulk. For creep processes, diﬀusion has to take place in both sublattices, but it

is controlled by the slower moving species, that is, the cations. Tracer diﬀusion

measurements yielded an activation energy of 4.7 eV [447]. According to the

dislocation loop shrinkage study in the lower temperature range between 1,100

and 1,300

◦

C, the activation energy (of climb) is 5.3 eV [443]. Certainly, the

latter study ﬁts best the situation of the present experiments. Thus, both the

stress exponent and the activation energy of the deformation around 1,400

◦

are close to the values suggested theoretically for recovery-controlled deforma-

tion. While the activation energy of cation diﬀusion seems to be independent

of the stabilizer ions and concentration, the diﬀusion coeﬃcient decreases by

more than one order of magnitude by increasing the yttria concentration from

7.3 Zirconia Single Crystals 273

about 10 to about 20 mol% [443,447]. This dependence explains the increased

ﬂow stress of the materials with higher yttria concentrations where recovery

is suppressed. At high temperatures, ZrO

–10 mol% Y

exhibits equal ﬂow

stresses at the diﬀerent loading axes as recovery eﬀects are independent of the

loading axis.

7.3.6 Summary of Cubic ZrO

At all temperatures, part of the ﬂow stress is of athermal nature due to long-

range interactions between dislocations and to the back stress of segments

bowing out between large jogs or dislocation nodes. This stress part controls

the athermal deformation of ZrO

–10 mol% Y

along 112around 1,000

◦

In all the other situations with higher ﬂow stresses, other mechanisms are

superimposed.

At the lowest temperatures where data are available, the Peierls mecha-

nism causes a steep increase of the ﬂow stress and the strain rate sensitivity

for glide on both the cube slip planes (activated in the 112 orientation) and

the non-cube slip planes (activated in the 100 orientation).

Although solution hardening by the yttrium ions may theoretically yield

a large contribution to the ﬂow stress, the temperature range between 500

and 800

◦

C with activation volumes up to more than 100 b

and dislocations

pinned between obstacles about 130 nm apart can best be interpreted by the

action of precipitation hardening. According to equal activation volumes of

the materials with diﬀerent yttria contents deformed along the soft 112

orientation, this mechanism does not cause the dependence of the ﬂow stress

on the yttria concentration at intermediate temperatures. It should, however,

be responsible for the higher ﬂow stress and the smaller activation volume of

non-cube slip at the deformation along 100.

The formation of yttrium ion atmospheres in the dislocation cores may

give rise to the higher ﬂow stress of the materials with higher yttria contents

due to strain ageing with serrated yielding at temperatures around 1,000

◦

Around 1,400

◦

C, deformation is controlled by recovery. As the diﬀusional

recovery processes are little inﬂuenced by the slip system, the ﬂow stress is

equal for the deformation of ZrO

–10 mol% Y

along 112 and 100.On

the other hand, the decrease of the cation diﬀusion coeﬃcient with increasing

yttria concentration results in the higher ﬂow stress of the materials with

higher yttria contents at 1,400

◦

C owing to reduced recovery.

7.3.7 Tetragonal ZrO

As outlined in the introduction and in Sect. 7.3.1, materials containing the

tetragonal phase of zirconia form a variety of microstructures.

274 7 Ceramic Single Crystals

[100]

[010]

Fig. 7.40. Tetragonal domains with their c axes in diﬀerent orientation forming

two colonies of diﬀerent composition of domains. The upper colony consists of the

variants t

and t

,thelower one of t

and t

Microstructure

The tetragonal phase of the diﬀerent microstructures consists of individual

domains with their c axes parallel to the three 100 axes of the corresponding

pseudo-cubic lattice. The domains are designated t

, t

,andt

. Domains of

alternating c axes are stacked to form colonies (e.g., [429]) as outlined in the

schematic picture of Fig. 7.40. The domain boundaries are the {101} twin

planes. The colonies grow preferentially in 111 directions. Owing to the

small deviations from orthogonality of the c axes, always three colonies of

equal growth direction, consisting of the diﬀerent combinations of the t

, t

and t

domains, form helical arrangements, where the boundaries between

the individual colonies are again {101} twin planes [448]. The colonies in the

metastable t



zirconia are supposed to have a structure similar to that of the

tetragonal precipitates in partially stabilized zirconia. In the 001 projection

of Fig. 7.40, the domain boundaries of the upper colony are inclined by 45

◦

with respect to the image plane, while in the lower colony they are oriented

edge-on. Respective micrographs are presented in Fig. 7.41.

Ferroelastic Deformation

Under stress, the tetragonal phase may undergo ferroelastic deformation [427–

429, 433,450]. During tensile loading along a pseudo-cubic axis, domains of c

axes diﬀerent from the loading axis may switch into that type with the c axis

in the tensile direction, forming a tetragonal single crystal. In compression,

that type of domains with the c axis parallel to the compression axis may

disappear. Domain switching in t



-ZrO

recorded during in situ straining in

an HVEM is illustrated in the following video clip.

Video 7.6. Domain switching in t



-ZrO

during ferroelastic deformation at

1, 150

◦

C: The video presents two short sequences of domain switching. The untrans-

formed areas appear in strong contrast owing to the electron microscopy contrast

7.3 Zirconia Single Crystals 275

1µm

(a)

(b)

Fig. 7.41. Domain and colony structure in t



-ZrO

. The edges of the images are

parallel to the cube axes. Domain boundaries inclined with respect to the image

plane and colony boundaries edge-on (a), colony borders inclined (b) corresponding

to the lower colony in Fig. 7.40 rotated by 90

◦

. From the work in [449]

of the domain boundaries and the orientation diﬀerences between the domains. The

contrast disappears during ferroelastic deformation when tetragonal single crystal

regions are formed. The transformation front moves preferentially in the direction

of the extension of the colonies rather than by the sidewise growth of the width of

the latter. Frame-by-frame analysis reveals that the individual domains transform

mostly in time intervals below the resolution of the video recording. This points at

an athermal character of the transformation process.

A partially transformed structure is also shown in Fig. 7.42. If the transforma-

tion starts simultaneously at diﬀerent points, the individual transformation

fronts will meet leaving behind antiphase boundary-like faults in the oxygen

sublattice. These residual defects are frequently located at the former colony

boundaries.

During compression, the transformation occurs in such a way that the

domains with their c axes parallel to the compression direction always assume

the character of their neighboring domains in the same colony. Thus, the

neighboring domains become tetragonal single crystal regions, and the high-

energy pseudotwin boundaries, where a c direction in one domain has to join

an a direction of another one, are generated only along the narrow areas along

the former colony boundaries.

The macroscopic behavior of t



-ZrO

during compressive loading along a

pseudo-cubic axis is demonstrated by the stress–strain curve in Fig. 7.43. The

elastic range ﬁnishes at a well deﬁned coercive stress σ

. There follows a range

of constant stress where the ferroelastic transformation takes place. As the

tetragonality of t



-ZrO

amounts to only about 1%, the maximum ferroelastic

strain in compression is about 1/3%. After the ferroelastic range, a second

276 7 Ceramic Single Crystals

1 µm

Fig. 7.42. Partially transformed colony structure in t



-ZrO

after in situ straining

in an HVEM at 1,150

◦

C. From the work in [450]

Fig. 7.43. Stress–strain curve of ferroelastic and plastic deformation of t



zirconia

at 500

◦

C. ε

total strain, Δε

transformation strain, σ

coercive stress, σ

plastic

yield stress, R

...R

stress relaxation tests. Data from [449]

elastic range appears followed by plastic deformation. Below about 1,200

◦

the coercive stress is lower than the yield stress of cubic ZrO

in the same

orientation. At 1,400

◦

C, it is higher. The strain rate sensitivity of the coercive

stress is always lower than that of the ﬂow stress of cubic zirconia underlining

the athermal character of the transformation.

Plastic Deformation

Dislocations with 1/2110 Burgers vectors as in the cubic phase are complete

dislocations in the tetragonal phase only if the Burgers vectors lie in the basal

plane. Otherwise, they are partial dislocations that should trail stacking faults

in the oxygen sublattice with antiphase boundary-like electron microscopy

7.3 Zirconia Single Crystals 277

contrast. Dislocations crossing a domain boundary should create a dislocation

in the boundary if their Burgers vector is not parallel to the boundary to fulﬁl

the Burgers vector conservation rule.

As discussed in [449], the microstructure after ferroelastic compressive

deformation consists of continuous regions of one tetragonal variant with their

c axis perpendicular to the compression axis, in which domains of the size of

the former colonies of the respective other variant are embedded. Figure 7.44

shows such regions in bright and dark contrast. After plastic deformation

parallel to the c axis of t

, two sets of dislocations are formed with Burgers

vectors parallel to [011] and [01

1] as illustrated in Fig. 7.44. The dislocations

0.5 μm

022

0.5 μm

02-2

Fig. 7.44. Microstructure in t



-ZrO

after ferroelastic and plastic deformation at

1,150

◦

C up to 1.3% under two g vectors near the [100] pole. The white bar marks

the compression direction. From [449]. Copyright Amer. Ceram. Soc. (2001)

278 7 Ceramic Single Crystals

probably extend on {111} planes and are strongly bowed-out with segment

lengths of the order of 0.1–0.15μm. Dislocations of diﬀerent activated slip

systems pin each other by mutual intersections.

The plastic yield stress of t



-ZrO

is very high at low temperatures (almost

1,400 MPa at 500

◦

C), decreasing down to about 650 MPa at 1,400

◦

C. Except

at 500

◦

C, the strain rate sensitivity is low (10 MPa). The ﬂow stress is inter-

preted by the sum of the long-range interaction between parallel dislocations

according to (7.13) with a dislocation density of 5 × 10

−2

,andtheback

stress of the segments bowing out between dislocation intersections given by

the Frank–Read stress (3.47). In agreement with the low strain rate sensitiv-

ity, the deformation is mainly of athermal nature. The high ﬂow stress at high

temperatures and the accompanying low strain rate sensitivity are explained

by recovery being prevented [449].

Partially Stabilized Zirconia

Manifold microstructures may form in partially stabilized zirconia depending

on the stabilizer concentration and the thermal treatment. Two structures are

shown in Fig. 7.45. The ﬁrst one consists of medium-sized platelet-like colonies

of only two t domain variants of approximately equal thickness embedded in

a matrix, which is supposed to be cubic. The other one contains few relatively

large precipitation colonies and a high density of small precipitates regularly

arranged in a so-called tweed structure. Both elements of the microstructure

have a similar morphology on diﬀerent size scales.

In situ straining experiments in an HVEM on the material of Fig. 7.45b

proved that ferroelastic deformation occurs also in partially stabilized zirconia.

In general, the untransformed regions show dark contrast under the imaging

conditions applied. The ﬁgure taken during in situ straining shows these dark

untransformed regions and brighter ones of the tweed structure which had

already been transformed. The following video demonstrates the process of

transformation for both the tweed structure and for individual domains in

larger colonies.

Video 7.7. Domain switching in partially stabilized ZrO

during ferroelastic defor-

mation at 1, 150

◦

C: The video clip contains three sequences of domain switching.

The ﬁrst one shows the partly transformed tweed structure and a few large pre-

cipitates with distinct domains. During straining the untransformed dark regions

shrink by the motion of the well deﬁned borders between untransformed and trans-

formed regions. Besides, some domains of the large precipitates transform after label

A. The second sequence presents again the shrinkage of the untransformed regions

with small precipitates, and the third one illustrates the switching of a large domain.

Consequently, in this material under tension the dislocations move after

ferroelastic deformation in a crystal that essentially is a tetragonal single

crystal.

Video 7.8. Dislocation motion in partially stabilized ZrO

after ferroelastic defor-

mation at 1, 150

◦

C: After partial transformation the dislocations move in the

7.3 Zirconia Single Crystals 279

1 µm

(a)

1 µm

(b)

Fig. 7.45. Microstructure in partially stabilized zirconia. Medium-sized t-ZrO

pre-

cipitate colonies in a cubic matrix (a), tweed structure of small precipitates and large

domains and colonies (b). From the work in [451] (a) and [433] (b)

transformed regions, in the second sequence of the video near larger precipitates.

The motion is quite jerky pointing at athermal dislocation motion as in t



zirconia.

Models of the strength of partially stabilized zirconia have been developed

on the basis of grain boundary hardening. They are reviewed in [433]. These

models do not consider the ferroelastic transformation, which may precede

280 7 Ceramic Single Crystals

plastic deformation. However, ferroelastic deformation has not yet been proved

in partially stabilized zirconia with larger precipitates like those in Fig. 7.45a.

Summary of Tetragonal Zirconia

The tetragonal phase of zirconia materials consists of domains of always two

of the three tetragonal variants in regular stacking to form larger colonies.

In t



zirconia and in some microstructures of partially stabilized zirconia,

ferroelastic domain switching precedes plastic deformation. As a result, the

original domain structure is destroyed, with tetragonal single crystal regions

with residual defects being formed, having at least the size of the former

colonies. It is not yet clear whether ferroelastic transformation takes place in

all microstructures of partially stabilized zirconia, or not.

During subsequent plastic deformation, the dislocations move in an ather-

mal way. The very high yield stress at high temperatures, which is a desired

design property, results from the suppression of recovery.

Metallic Alloys

The alloys described in this chapter are hardened either by precipitates or

by extrinsic particles. As the concentration and size of these obstacles are

low, the dislocation motion can be described by the theory of localized obsta-

cles (Sect. 4.5). In the special case where the hardening mechanism inside

the particles consists in the formation of antiphase boundaries, as in Al–Li,

the matrix dislocations may move in pairs. Impenetrable obstacles like oxide

dispersoids are circumvented by the formation of Orowan loops at low temper-

atures (Sect. 4.7). At high temperatures, they are overcome by climb. However,

there may be an attractive interaction between the particles and the disloca-

tions, resulting in a threshold stress for plastic deformation even at relatively

high temperatures.

8.1 Precipitation Hardened Aluminium Alloys

Aluminium has the f.c.c. structure with {111}1/2110slip systems as treated

in detail in Sect. 3.3. Pure aluminium is very soft at room temperature as

the Peierls stress in these close packed metals is important only at very low

temperatures. For practical application, aluminium-base alloys are mostly

hardened by precipitate particles of other phases (for a review see Nem-

bach [225]). Aluminium alloys are frequently solution treated, that is, they

are heated for a few hours at about 500

◦

C to solve the additions, followed

by quenching down to room temperature. Afterwards, they are aged at tem-

peratures between room temperature and up to some 200

◦

C to precipitate

the particles. During ageing, the particle diameter grows following a t

1/3

rule,

where t is the ageing time. The large particles grow at the expense of the

smaller ones. Thus, the concentration of particles decreases during ageing. As

the particle strength no longer increases when the particles are surmounted by

Orowan looping, the yield stress experiences a maximum at a certain ageing

time. This state is called peak-aged. Alloys with lower yield stress owing to